From 9ee76e70213b3b29fb712e9003ea610b6c446579 Mon Sep 17 00:00:00 2001 From: Arshan Dalili <57400500+arshandalili@users.noreply.github.com> Date: Fri, 17 Dec 2021 23:56:46 +0330 Subject: [PATCH 01/33] Update index.yml --- notebooks/index.yml | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/notebooks/index.yml b/notebooks/index.yml index 1b2133c0..d4ccffd3 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -54,4 +54,6 @@ notebooks: kind: S2021, LN, Notebook #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ - kind: S2021, LN, Notebook \ No newline at end of file + kind: S2021, LN, Notebook + - notebook: notebooks/12_temporal_probability_models/ + kind: S2021, LN, Notebook From e7dabe0064f1a7958c7c133637a96df2f520d640 Mon Sep 17 00:00:00 2001 From: Arshan Date: Fri, 17 Dec 2021 23:58:13 +0330 Subject: [PATCH 02/33] 12 --- .../12_temporal_probability_models/index.md | 390 ++++++++++++++++++ .../metadata.yml | 25 ++ .../resources/9tlhpJ9.png | Bin 0 -> 262076 bytes .../resources/N6fC8TW.png | Bin 0 -> 27644 bytes .../resources/WmoLVeV.png | Bin 0 -> 255712 bytes .../resources/YOld6zJ.jpeg | Bin 0 -> 58111 bytes .../automatic-speech-recognition_updated.png | Bin 0 -> 95828 bytes .../resources/k4JYqIC.png | Bin 0 -> 36763 bytes ...tutorial_illustration_v3-02-3000x1351.jpeg | Bin 0 -> 100705 bytes ...tutorial_illustration_v3_02_3000x1351.jpeg | Bin 0 -> 100705 bytes .../resources/lnpogDU.png | Bin 0 -> 245271 bytes .../resources/sFAP5bQ.jpeg | Bin 0 -> 82945 bytes .../resources/textra.png | Bin 0 -> 5964 bytes .../resources/wvnlp2L.jpeg | Bin 0 -> 157699 bytes 14 files changed, 415 insertions(+) create mode 100644 notebooks/12_temporal_probability_models/index.md create mode 100644 notebooks/12_temporal_probability_models/metadata.yml create mode 100644 notebooks/12_temporal_probability_models/resources/9tlhpJ9.png create mode 100644 notebooks/12_temporal_probability_models/resources/N6fC8TW.png create mode 100644 notebooks/12_temporal_probability_models/resources/WmoLVeV.png create mode 100644 notebooks/12_temporal_probability_models/resources/YOld6zJ.jpeg create mode 100644 notebooks/12_temporal_probability_models/resources/automatic-speech-recognition_updated.png create mode 100644 notebooks/12_temporal_probability_models/resources/k4JYqIC.png create mode 100644 notebooks/12_temporal_probability_models/resources/kapernikov_robot_localization_tutorial_illustration_v3-02-3000x1351.jpeg create mode 100644 notebooks/12_temporal_probability_models/resources/kapernikov_robot_localization_tutorial_illustration_v3_02_3000x1351.jpeg create mode 100644 notebooks/12_temporal_probability_models/resources/lnpogDU.png create mode 100644 notebooks/12_temporal_probability_models/resources/sFAP5bQ.jpeg create mode 100644 notebooks/12_temporal_probability_models/resources/textra.png create mode 100644 notebooks/12_temporal_probability_models/resources/wvnlp2L.jpeg diff --git a/notebooks/12_temporal_probability_models/index.md b/notebooks/12_temporal_probability_models/index.md new file mode 100644 index 00000000..a0d822a8 --- /dev/null +++ b/notebooks/12_temporal_probability_models/index.md @@ -0,0 +1,390 @@ +# Temporal Probability Models +## Group Info: +Group Number: 12 +Members: Arshan Dalili, Soroush Jahanzad, Mohammad Hejri + + + + + +## Table of Contents +[Introduction](#Introduction) + +[Reasoning over time or space](#Reasoning-over-time-or-space) +* [What is a Temporal Probability Model?](#What-is-a-Temporal-Probability-Model) +* [Markov Models](#Markov-Models) +* [Joint Distributions of Markov Models](#Joint-Distributions-of-Markov-Models) +* [Chain Rule](#Chain-Rule) + +[Mini-Forward Algorithm](#Mini-Forward-Algorithm) + +[Stationary Distribution](#Stationary-Distribution) + +[Hidden Markov Model](#Hidden-Markov-Model) +* [Chain Rule and Joint Distribution of Hidden Markov Model](#Chain-Rule-and-Joint-Distribution-of-Hidden-Markov-Model) +* [Hidden Markov Models Applications](#Hidden-Markov-Model-Applications) + +[Conclusion](#Conclusion) + +[References](#References) + + + + + +## Introduction +In this lecture note, we are going to talk about **_Temporal Probability Models_**. We will discuss what they are and how they can help us model problems that are subject to change over time. + + +## Reasoning over time or space +So far, we have learned how to deal with probabilistic reasoning in static worlds. That means each random variable has a fixed value and doesn't change over time. However, most of our problems don't have this static behavior, and the state varies over time (or space). + +Let' start with an example of a static problem: +>Imagine that your car has broken down. You may guess that the problem is related to the engine, batteries, or other car parts. You may take the vehicle to a car repair shop to diagnose the problem and fix it. You know that during the diagnosis process, the broken part remains broken, so the state of the car doesn't change. +
+car repair +
+ + +But our problems aren't always static. Consider the below example: +>Take diabetes management of a patient as our problem. We know that, for instance, the patient's blood sugar isn't constant and will vary over time, so there is a dynamic in the variables used to describe the patient's state (such as the blood sugar and insulin). + + +
+diabetes +
+ + +As you can see, the patient's status varies over time, and our approaches in static probabilistic reasoning won't work for these problems. So we should look for other approaches to solve them. + +Temporal Probability Models can make it easier for us to model and solve these types of problems. + + +### What is a Temporal Probability Model? + +As the diabetes example above shows, we need to introduce **time** (or space) for our models. +Retake variables of the diabetes example. We know that these variables can change over time, so to manage the diabetes of the patient, we slice time at specific points and keep track of this evidence to evaluate the current state and predict the future diabetic condition of the patient. + +>_**Hence we can model the diabetes status of a patient using a Temporal Probability Model.**_ + +There are some variables like measured blood sugar and pulse rate that are measurable, and we denote them as ${E}_t$ at time $t$. + +On the other hand, we have some unmeasurable variables like stomach content and blood sugar. (Notice that measured blood sugar and blood sugar are considered different variables since measured blood sugar has some errors.) and at time $t$, we use ${X}_t$ to denote them. +In a nutshell, we have: + +\begin{equation} +{E}_t = Observable\;Variable\;at\;time\;t\\ +{X}_t = NonObservable\;Variable\;at\;time\;t +\end{equation} + + + +### Markov Models +Let's define a random variable $X$ and call its value at a given time $t$ the state of $X$ and denote it as $X_t$. If in our model the future state of $X$ depends only on the current state of $X$ and not on states that had occurred before, we call it a **_First-order Markov Model_**, or simply a **_Markov Model_**. Bayesian network of this model looks like the below network and is referred to as a **_Markov Chain_**: +
+Bayes Net of Markov Model +
+ +In a Markov chain, we can write the Morkov assumption as: + +\begin{equation} +P(X_n=x_n|X_{1:n-1}=x_{1:n-1}) = P(X_n = x_n|X_{n - 1}=x_{n - 1}) +\end{equation} + +*Note:* We also define $X_{a:b}$ to denote a set of variables from $X_a$ to $X_b$. + +\begin{equation} +X_{a:b} = X_a, X_{a+1}, \cdots, X_{b-1}, X_b +\end{equation} + + +In a similar way, we can use **_$i$th-order Markov Models_** to model situations in which the future state of $X$ depends only on $i$ recent states and not on states that had occrred before. In this case we have: + +\begin{equation} +P(X_n=x_n|X_{1:n-1}=x_{1:n-1}) = P(X_n = x|X_{n - i:n - 1}=x_{n - i : n - 1}) +\end{equation} + +### Joint Distributions of Markov Models + +Take $X$ as a Markov Model and consider its Bayesian network (depicted below). +
+Bayes Net of Markov Model +
+ +We can quickly notice that for every $1\lt i\le n$ , if we are given $X_{i-1}$, then **$X_i$ is independent of $X_1, X_2, \cdots, X_{i-2}$** since there is only one inactive path between each of them and $X_i$. (Regarding the fact that $X_{i-1}$ is given) +So we have: + +\begin{equation} + X_i \perp \!\!\! \perp X_1, X_2, \cdots, X_{i-2} \vert X_{i-1} +\end{equation} + +We define two parameters here: +* **Transition Probabilities**: + + \begin{equation} + P(X_t \vert X_{t-1}) + \end{equation} + + It shows how each state is related to its previous state and how states develop. + The **_Stationary assumption_** in Markov models ensures that transition probabilities don't change over time. +* **Initial State Probability**: + + \begin{equation} + P(X_1) + \end{equation} + + It is the probability of the initial state ($X_1$), and we usually know its value. + + We can find the joint distribution of X regarding the Bayesian network and the parameters we defined. If we consider $X_1$ to $X_n$, then we have: + + \begin{equation} + P(X_1, X_2, \cdots, X_{n-1}, X_n) = P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n-1}\vert X_{n-2})P(X_{n}\vert X_{n-1})\\ \Rightarrow P(X_1, X_2, \cdots, X_{n-1}, X_n) =P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) + \end{equation} + +In the next section, we will prove this joint distribution using Chain Rule and independence. + + + + +### Chain Rule +Take $X$ as a Markov Model and consider its Bayesian network (depicted below). +
+Bayes Net of Markov Model +
+ +We know that from Chain Rule, every joint distribution of $X_1$ to $X_n$ can be written as: + + \begin{equation} + P(X_1, X_2, \cdots, X_{n-1}, X_n) = P(X_1)P(X_2\vert X_1)P(X_3\vert X_1, X_2)\cdots P(X_{n}\vert X_1,X_2,\cdots X_{n-1}) + \end{equation} + +Now, due to the independence we saw earlier, we can assume: + +\begin{equation} + X_3 \perp \!\!\! \perp X_1\vert X_2, X_4\perp \!\!\! \perp X_1,X_2\vert X_3, \cdots, X_{n}\perp \!\!\! \perp X_1,X_2\cdots X_{n-2}\vert X_{n-1} +\end{equation} + +So, after applying independence to Chain Rule, we have: + + \begin{equation} + P(X_1, X_2, \cdots, X_{n-1}, X_n)= P(X_1)P(X_2\vert X_1)P(X_3\vert X_1, X_2)\cdots P(X_{n}\vert X_1,X_2,\cdots X_{n-1})\\=P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n}\vert X_{n-1})\\=P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) + \end{equation} + +which is the same result we get from the previous section. + +## Mini-Forward Algorithm + +Imagine you want to go on a trip with your friends. Your friends think that trips in sunny days are more fun, so they ask you to schedule the trip for a sunny day. You know that you cannot be completely certain whether the $n$th day after today is going to be sunny or rainy, so what will you do? + +We can solve this problem easily if we assume a Markov model. Suppose that we have measured the transition probabilities for this model and know that if some day is sunny, the next day will be sunny with probability $P(X_{n + 1} = sunny | X_{n} = sunny) = 0.9$ and will be rainy otherwise. Similarly, if some day is rainy, the next day will be rainy with probability $P(X_{n + 1} = rainy | X_{n} = rainy) = 0.7$ and will be sunny otherwise. We also know the initial probability because we know today's weather. + +Now that we have a good model, we need an efficient approach to find the probability of each outcome in the $n$th day after today. A naive approach would be to enumerate all possible sequences of length $n$ which end in each outcome and then add up their probabilities. This will do the work, but it will require a lot of time and effort and is too slow. A better approach is to store our incremental *belief* about the weather and to update it as we consider proceed twords the $n$th day in our calculations. This approach is called the **Mini-Forward Algorithm**. + +In the Mini-Forward algorithm, we use the probabilities for the initial state and forward simulation to calculate the probabilities in the future states. The probability of outcome $x_i$ in the state $X_n$ using this algorithm is: + +\begin{equation} + P(X_n = x_i) = \sum_{x_j} P(X_n = x_i | X_{n - 1} = x_j) P(X_{n - 1} = x_j) +\end{equation} + +This can be easily computed for each state because we know the initial probability $P(X_1 = x_k)$ for every $k$ and the transition probabilities for each state can also be calculated using this algorothm. + +Now that we have this solution, we can say that the probability that the $n$th day after today will be sunny in our example is: + +\begin{equation} + P(X_{n + 1} = sunny) = \sum_{x_j \in \{sunny, rainy\}} P(X_n = sunny | X_{n - 1} = x_j) P(X_{n - 1} = x_j) +\end{equation} + +\begin{equation} + \Rightarrow P(X_{n + 1} = sunny) = 0.9 \times P(X_n = sunny | X_{n - 1} = sunny) + 0.3 \times P(X_n = sunny | X_{n - 1} = rainy) +\end{equation} + +Now, let's take a look at the state of weather with initial observation of sun: +
+Initial Sunny Day +
+
+ +Finally, let's take a look at the state of weather with initial observation of rain: + +
+Initial Sunny Day +
+ + + +## Stationary Distribution + +An interesting question to ask in problems modeled using Markov chains is about the state as we continue simulating for a long time. As we simulate further into the future, the Moarkov chain becomes longer and uncertainty is accumulated. + +In the general case, we cannot say what the state will be in the far future. However, for most Markov chains, we will eventually end up in the same distribution no matter what the initial distribution is. In other words, the distribution we end up in is independent of the initial distribution. In such cases, the distribution we end up with is called the **stationary distribution** of the chain. This distribution is denoted by $P_\infty$ and satisfies the condition below. + +\begin{equation} + P_{\infty}(X) = P_{\infty + 1}(X) = \sum_{x} P(X|x) P_{\infty}(x) +\end{equation} + +In the weather example we mentioned earlier, the stationary distribution denotes the probability that each day will be sunny or rainy in general. + + +## Most Likely Explanation: Mini-Viterbi Algorithm + +Sometimes we would like to find out what happened in the past using the knowledge that we have in present. Consider the trip we discussed earlier as an example. Imagine that you and your friends want to go outside, but some of your friends warn you that everything might be wet if it had rained in the past $n$ days. Since your destination is too far, you do not know about its past, so we need to reason about the most likely sequence of your destination's weather in the past $n$ days. The course of this problem can be shown using the following figure. + +
+Weather CPT +
+ +Again, a slow answer would be to enumerate all possible combinations and calculate the probabilities. However, we can calculate the most likely sequence inductively, remember the incremental probabilities, and update the most likely sequence as we proceed. This approach is called the **Mini-Viterbi Algorithm**. + +In the Mini-Viterbi algorithm, we define $m_t[x]$ and $a_t[x]$ as: + +\begin{equation} + m_t[x] = \max_{x_{1:t - 1}} P(x_{1:t - 1}, x) +\end{equation} +\begin{equation} + a_t[x] = arg\!\max_{x_{1:t - 1}} P(x_{1:t - 1}, x) +\end{equation} + +The equation above can be rewritten to: +\begin{equation} + m_1[x] = P(X_1 = x1) +\end{equation} +\begin{equation} + m_t[x] = \max_{x_{1:t - 1}} P(x_t | x_{t - 1}) m_{t - 1}[x] +\end{equation} + +Using this method, the most likely sequence in the past $t - 1$ states, also known as the most likely explanation, will be available in $a_t[x]$. + +In our example, we can use the Mini-Viterbi algorithm starting from $x_1 = The\ weather\ n\ days\ ago$, and the most likely sequence would be available in $a_{n + 1}[x_{today}]$. + +## Hidden Markov Model +Let's start with the following scenario: + +We have two friends Alice and Bob. They live very far apart. So, they talk on the phone. Bob has a mood that changes based on the weather. Whenever they talk on the phone, Bob tells Alice his mood. + +Suppose that if the weather is sunny, then Bob is happy with a probability of 80% and grumpy with a probability of 20%. Likewise, if the weather is rainy, Bob is grumpy with a probability of 60% and happy with a probability of 40%. + +Whenever Bob tells Alice that he is happy, she may guess that the weather is sunny regarding the probabilities discussed above. Similarly, if Bob tells Alice that he is grumpy, she may suppose rainy weather. Notice that there may be some exceptions that despite the rainy weather, Bob is happy, or despite the sunny weather, he is grumpy. + +Now, we get the time involved in our problem. Suppose that if today is sunny, tomorrow will be sunny with the probability of 80%, and rainy with the probability of 20%. Likewise, if today is rainy, tomorrow will be rainy with the probability of 60%, and sunny with the probability of 40%. + +Consider the figure below to get a better understanding of the problem discussed above: +
+Bayes Net of Markov Model +
+ +Now, we formulate the problem formally: + +\begin{equation} +{E}_t: Bob's\;mood\;at\;time\;t\;(Observation) \in \{happy, grumpy\}\\ +{X}_t: Weather's\;condition\;at\;time\;t\;(State) \in \{sunny, rainy\} +\end{equation} +So, the question is, how can Alice infer the weather’s condition (state) regarding Bob’s mood (observation)? + +The problem discussed above is an example of the Hidden Markov Model (HMM). It is not possible to model the problem as a Markov Chain in many problems. So, how can we model these problems? + +Most of the time, it is possible to use HMM! If the problem has a superior state that is a Markov Process, we can make an assumption about this superior state (i.e. our belief), and update it as we observe the problem’s inputs. + +In the problem discussed before, our belief is weather’s condition that can be updated regarding the observation, Bob’s mood. + +Let’s get into HMM’s structure. We start with a simple Markov chain. At each state, we add new nodes for the inputs of the problem. These nodes are called observation nodes and act like agent sensors. Note that each observation node solely relies on its corresponding state. + +Take a closer look at the figure below to find out HMM's structure: +
+Bayes Net of Markov Model +
+
+ +Regarding the structure above, an HMM is defined by: +- $P(X_1)$ as initial distribution +- $P(X_t \mid X_{t-1})$ as transitions distribution +- $P(E_t \mid X_t)$ as emissions distribution + +We will dive into more mathematics in the next part. + +### Chain Rule and Joint Distribution of Hidden Markov Model +Let's denote $X_t,X_{t-1},\cdots,X_2,X_1$ by $X_{t:1}$. Similarly, denote $E_t,E_{t-1},\cdots,E_2,E_1$ by $E_{t:1}$. + +We are going to calculate joint distribution of HMM. + +From the Chain Rule, every joint distribution over $X_{t:1}, E_{t:1}$ can be written as: + +\begin{equation} +P(X_{t:1}, E_{t:1})=P(X_1)P(E_1 \mid X_1)P(X_2\mid X_1,E_1)P(E_2\mid X_{2:1},E_1)\cdots P(X_t\mid X_{t-1:1},E_{t-1:1})P(E_t\mid X_{t:1},E_{t-1:1}) +\end{equation} +Regarding the HMM's structure, we can assume that: + +\begin{equation} +\forall t>1:\;\;\;X_t \perp \!\!\! \perp X_{t-2:1},E_{t-1:1} \mid X_{t-1}\;\;\;and\;\;\;E_t \perp \!\!\! \perp X_{t-1:1},E_{t-1:1} \mid X_{t} +\end{equation} +Recall that every node given its parents is independent of its non-descendants. + +Using the properties above, we get: +\begin{equation} +P(X_{t:1}, E_{t:1})=P(X_1)P(E_1 \mid X_1)\prod_{i = 2}^{t} P(X_i \mid X_{i-1})P(E_i \mid X_i) +\end{equation} + +### Hidden Markov Model Applications +Hidden Markov models are known for their applications in various fields, such as: +- Signal processing +- Information theory +- Pattern recognition +- Statistical mechanics +- Physics +- Chemistry +- Economics +- Finance +- Thermodynamics + + + +Below, some of its popular applications are described in more details: + +#### Speech recognition HMMs +The main core of HMM-based speech recognition systems is the Viterbi algorithm which will be discussed in the following section. The Viterbi algorithm uses dynamic programming to find out the best alignment between the input speech and a given speech model. Nowadays, most Large Vocabulary Continuous Speech Recognition (LVCSR) systems are based on HMMs. + +In speech recognition HMMs, the continuous-valued acoustic signals are our observations. States are particular positions in specific words. + +
+speech recognition +
+ + + +#### Machine translation HMMs +A simple approach would be the mechanical substitution of words in one language for words in another. However, this won't result in coherent translations most of the time. + +Better results can be achieved using attention-based neural machine translation (NMT) models. These models focus on specific source positions to produce a translation with considerable improvements over pure encoder-decoder models. To achieve this, We use a Hidden Markov Model (HMM) consisting of neural network-based alignment and lexicon models, which are trained jointly using the forward-backward algorithm. + +In machine translation HMMs, observations are millions of words and corresponding states are translation options. + +
+machine translation +
+ +#### Robot tracking HMMs +Hidden Markov Models (HMM) are applied to interoceptive data acquired by a moving robot when contouring an environment. To track the robot accurately, the robot needs a sufficient set of sensors. + +In Robot tracking HMMs, observations are sensors' continuous perceptions and corresponding states are positions in the map. + +
+machine translation +
+ +## Conclusion +In this note, we first introduced Temporal Probability Model and discussed why we need them to model problems that vary over time. We then introduce Markov Models, explain their properties, and calculate their joint distribution with Chain Rule. + +Then, we talked about the Mini-Forward algorithm and how it can help us predict the state of the system in the future. After that, we talked about stationary distributions help us to know will eventually end up in the same distribution no matter what the initial distribution is. + +Then, we discussed a popular family of temporal models, called Hidden Markov Models for probabilistic sequence classification. In these models, we link a sequence of observations to a sequence of hidden states that explains those observations. Inference, as our goal, was the process of discovering the sequence of hidden states, given the sequence +of observations. Finally, we introduced some of the HMM's well-known applications. + +## References +* [Artificial Intelligence: A Modern Approach, 4th Edition](http://aima.cs.berkeley.edu/) +* [CE417 Lecture Slides, Prof. Rohabn, Sharif University of Technology](http://ce.sharif.edu/courses/00-01/1/ce417-1/resources/root/Slides/PDF/Session%2015_16.pdf) +* [CSC545 Lecture Slides, University of Miami](https://www.cs.miami.edu/home/visser/csc545-files/ProbabilisticReasoningOverTime.pdf) +* [Speech and Language Processing. Daniel Jurafsky & James H. Martin, Stanford University](https://web.stanford.edu/~jurafsky/slp3/A.pdf) +* [University of Tennesee Lecture Slides](https://nanopdf.com/download/chapter-15-probabilistic-reasoning-over-time_pdf) +* [Paulo E. Santos (FEI - Sao Paulo) Slides](https://fei.edu.br/~psantos/slidesIA/CAP15-DBN.pdf) +* [CS 188: Artificial Intelligence Lecture Slides, Dan Klein, Spring 2006](https://inst.eecs.berkeley.edu/~cs188/sp06/slides/cs188%20lecture%2018%20--%20HMMs%20(2pp).pdf) + +_\*Some conceptual pictures used in this note are under courtesy of [VectorStock Media](https://vectorstock.com)_ \ No newline at end of file diff --git a/notebooks/12_temporal_probability_models/metadata.yml b/notebooks/12_temporal_probability_models/metadata.yml new file mode 100644 index 00000000..544564f5 --- /dev/null +++ b/notebooks/12_temporal_probability_models/metadata.yml @@ -0,0 +1,25 @@ +title: # shown on browser tab + +header: + title: # title of your notebook + description: # short description of your notebook + +authors: + label: + position: top + content: + # list of notebook authors + - name: # name of author + role: Author # change this if you want + contact: + # list of contact information + - link: 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Algorithm](#Mini-Forward-Algorithm) + +[Stationary Distribution](#Stationary-Distribution) + +[Hidden Markov Model](#Hidden-Markov-Model) +* [Chain Rule and Joint Distribution of Hidden Markov Model](#Chain-Rule-and-Joint-Distribution-of-Hidden-Markov-Model) +* [Hidden Markov Models Applications](#Hidden-Markov-Model-Applications) + +[Conclusion](#Conclusion) + +[References](#References) + + + + + +## Introduction +In this lecture note, we are going to talk about **_Temporal Probability Models_**. We will discuss what they are and how they can help us model problems that are subject to change over time. + + +## Reasoning over time or space +So far, we have learned how to deal with probabilistic reasoning in static worlds. That means each random variable has a fixed value and doesn't change over time. However, most of our problems don't have this static behavior, and the state varies over time (or space). + +Let' start with an example of a static problem: +>Imagine that your car has broken down. You may guess that the problem is related to the engine, batteries, or other car parts. You may take the vehicle to a car repair shop to diagnose the problem and fix it. You know that during the diagnosis process, the broken part remains broken, so the state of the car doesn't change. +

+car repair +
+ + +But our problems aren't always static. Consider the below example: +>Take diabetes management of a patient as our problem. We know that, for instance, the patient's blood sugar isn't constant and will vary over time, so there is a dynamic in the variables used to describe the patient's state (such as the blood sugar and insulin). + + +
+diabetes +
+ + +As you can see, the patient's status varies over time, and our approaches in static probabilistic reasoning won't work for these problems. So we should look for other approaches to solve them. + +Temporal Probability Models can make it easier for us to model and solve these types of problems. + + +### What is a Temporal Probability Model? + +As the diabetes example above shows, we need to introduce **time** (or space) for our models. +Retake variables of the diabetes example. We know that these variables can change over time, so to manage the diabetes of the patient, we slice time at specific points and keep track of this evidence to evaluate the current state and predict the future diabetic condition of the patient. + +>_**Hence we can model the diabetes status of a patient using a Temporal Probability Model.**_ + +There are some variables like measured blood sugar and pulse rate that are measurable, and we denote them as ${E}_t$ at time $t$. + +On the other hand, we have some unmeasurable variables like stomach content and blood sugar. (Notice that measured blood sugar and blood sugar are considered different variables since measured blood sugar has some errors.) and at time $t$, we use ${X}_t$ to denote them. +In a nutshell, we have: + +\begin{equation} +{E}_t = Observable\;Variable\;at\;time\;t\\ +{X}_t = NonObservable\;Variable\;at\;time\;t +\end{equation} + + + +### Markov Models +Let's define a random variable $X$ and call its value at a given time $t$ the state of $X$ and denote it as $X_t$. If in our model the future state of $X$ depends only on the current state of $X$ and not on states that had occurred before, we call it a **_First-order Markov Model_**, or simply a **_Markov Model_**. Bayesian network of this model looks like the below network and is referred to as a **_Markov Chain_**: +
+Bayes Net of Markov Model +
+ +In a Markov chain, we can write the Morkov assumption as: + +\begin{equation} +P(X_n=x_n|X_{1:n-1}=x_{1:n-1}) = P(X_n = x_n|X_{n - 1}=x_{n - 1}) +\end{equation} + +*Note:* We also define $X_{a:b}$ to denote a set of variables from $X_a$ to $X_b$. + +\begin{equation} +X_{a:b} = X_a, X_{a+1}, \cdots, X_{b-1}, X_b +\end{equation} + + +In a similar way, we can use **_$i$th-order Markov Models_** to model situations in which the future state of $X$ depends only on $i$ recent states and not on states that had occrred before. In this case we have: + +\begin{equation} +P(X_n=x_n|X_{1:n-1}=x_{1:n-1}) = P(X_n = x|X_{n - i:n - 1}=x_{n - i : n - 1}) +\end{equation} + +### Joint Distributions of Markov Models + +Take $X$ as a Markov Model and consider its Bayesian network (depicted below). +
+Bayes Net of Markov Model +
+ +We can quickly notice that for every $1\lt i\le n$ , if we are given $X_{i-1}$, then **$X_i$ is independent of $X_1, X_2, \cdots, X_{i-2}$** since there is only one inactive path between each of them and $X_i$. (Regarding the fact that $X_{i-1}$ is given) +So we have: + +\begin{equation} + X_i \perp \!\!\! \perp X_1, X_2, \cdots, X_{i-2} \vert X_{i-1} +\end{equation} + +We define two parameters here: +* **Transition Probabilities**: + + \begin{equation} + P(X_t \vert X_{t-1}) + \end{equation} + + It shows how each state is related to its previous state and how states develop. + The **_Stationary assumption_** in Markov models ensures that transition probabilities don't change over time. +* **Initial State Probability**: + + \begin{equation} + P(X_1) + \end{equation} + + It is the probability of the initial state ($X_1$), and we usually know its value. + + We can find the joint distribution of X regarding the Bayesian network and the parameters we defined. If we consider $X_1$ to $X_n$, then we have: + + \begin{equation} + P(X_1, X_2, \cdots, X_{n-1}, X_n) = P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n-1}\vert X_{n-2})P(X_{n}\vert X_{n-1})\\ \Rightarrow P(X_1, X_2, \cdots, X_{n-1}, X_n) =P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) + \end{equation} + +In the next section, we will prove this joint distribution using Chain Rule and independence. + + + + +### Chain Rule +Take $X$ as a Markov Model and consider its Bayesian network (depicted below). +
+Bayes Net of Markov Model +
+ +We know that from Chain Rule, every joint distribution of $X_1$ to $X_n$ can be written as: + + \begin{equation} + P(X_1, X_2, \cdots, X_{n-1}, X_n) = P(X_1)P(X_2\vert X_1)P(X_3\vert X_1, X_2)\cdots P(X_{n}\vert X_1,X_2,\cdots X_{n-1}) + \end{equation} + +Now, due to the independence we saw earlier, we can assume: + +\begin{equation} + X_3 \perp \!\!\! \perp X_1\vert X_2, X_4\perp \!\!\! \perp X_1,X_2\vert X_3, \cdots, X_{n}\perp \!\!\! \perp X_1,X_2\cdots X_{n-2}\vert X_{n-1} +\end{equation} + +So, after applying independence to Chain Rule, we have: + + \begin{equation} + P(X_1, X_2, \cdots, X_{n-1}, X_n)= P(X_1)P(X_2\vert X_1)P(X_3\vert X_1, X_2)\cdots P(X_{n}\vert X_1,X_2,\cdots X_{n-1})\\=P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n}\vert X_{n-1})\\=P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) + \end{equation} + +which is the same result we get from the previous section. + +## Mini-Forward Algorithm + +Imagine you want to go on a trip with your friends. Your friends think that trips in sunny days are more fun, so they ask you to schedule the trip for a sunny day. You know that you cannot be completely certain whether the $n$th day after today is going to be sunny or rainy, so what will you do? + +We can solve this problem easily if we assume a Markov model. Suppose that we have measured the transition probabilities for this model and know that if some day is sunny, the next day will be sunny with probability $P(X_{n + 1} = sunny | X_{n} = sunny) = 0.9$ and will be rainy otherwise. Similarly, if some day is rainy, the next day will be rainy with probability $P(X_{n + 1} = rainy | X_{n} = rainy) = 0.7$ and will be sunny otherwise. We also know the initial probability because we know today's weather. + +Now that we have a good model, we need an efficient approach to find the probability of each outcome in the $n$th day after today. A naive approach would be to enumerate all possible sequences of length $n$ which end in each outcome and then add up their probabilities. This will do the work, but it will require a lot of time and effort and is too slow. A better approach is to store our incremental *belief* about the weather and to update it as we consider proceed twords the $n$th day in our calculations. This approach is called the **Mini-Forward Algorithm**. + +In the Mini-Forward algorithm, we use the probabilities for the initial state and forward simulation to calculate the probabilities in the future states. The probability of outcome $x_i$ in the state $X_n$ using this algorithm is: + +\begin{equation} + P(X_n = x_i) = \sum_{x_j} P(X_n = x_i | X_{n - 1} = x_j) P(X_{n - 1} = x_j) +\end{equation} + +This can be easily computed for each state because we know the initial probability $P(X_1 = x_k)$ for every $k$ and the transition probabilities for each state can also be calculated using this algorothm. + +Now that we have this solution, we can say that the probability that the $n$th day after today will be sunny in our example is: + +\begin{equation} + P(X_{n + 1} = sunny) = \sum_{x_j \in \{sunny, rainy\}} P(X_n = sunny | X_{n - 1} = x_j) P(X_{n - 1} = x_j) +\end{equation} + +\begin{equation} + \Rightarrow P(X_{n + 1} = sunny) = 0.9 \times P(X_n = sunny | X_{n - 1} = sunny) + 0.3 \times P(X_n = sunny | X_{n - 1} = rainy) +\end{equation} + +Now, let's take a look at the state of weather with initial observation of sun: +
+Initial Sunny Day +
+
+ +Finally, let's take a look at the state of weather with initial observation of rain: + +
+Initial Sunny Day +
+ + + +## Stationary Distribution + +An interesting question to ask in problems modeled using Markov chains is about the state as we continue simulating for a long time. As we simulate further into the future, the Moarkov chain becomes longer and uncertainty is accumulated. + +In the general case, we cannot say what the state will be in the far future. However, for most Markov chains, we will eventually end up in the same distribution no matter what the initial distribution is. In other words, the distribution we end up in is independent of the initial distribution. In such cases, the distribution we end up with is called the **stationary distribution** of the chain. This distribution is denoted by $P_\infty$ and satisfies the condition below. + +\begin{equation} + P_{\infty}(X) = P_{\infty + 1}(X) = \sum_{x} P(X|x) P_{\infty}(x) +\end{equation} + +In the weather example we mentioned earlier, the stationary distribution denotes the probability that each day will be sunny or rainy in general. + + +## Most Likely Explanation: Mini-Viterbi Algorithm + +Sometimes we would like to find out what happened in the past using the knowledge that we have in present. Consider the trip we discussed earlier as an example. Imagine that you and your friends want to go outside, but some of your friends warn you that everything might be wet if it had rained in the past $n$ days. Since your destination is too far, you do not know about its past, so we need to reason about the most likely sequence of your destination's weather in the past $n$ days. The course of this problem can be shown using the following figure. + +
+Weather CPT +
+ +Again, a slow answer would be to enumerate all possible combinations and calculate the probabilities. However, we can calculate the most likely sequence inductively, remember the incremental probabilities, and update the most likely sequence as we proceed. This approach is called the **Mini-Viterbi Algorithm**. + +In the Mini-Viterbi algorithm, we define $m_t[x]$ and $a_t[x]$ as: + +\begin{equation} + m_t[x] = \max_{x_{1:t - 1}} P(x_{1:t - 1}, x) +\end{equation} +\begin{equation} + a_t[x] = arg\!\max_{x_{1:t - 1}} P(x_{1:t - 1}, x) +\end{equation} + +The equation above can be rewritten to: +\begin{equation} + m_1[x] = P(X_1 = x1) +\end{equation} +\begin{equation} + m_t[x] = \max_{x_{1:t - 1}} P(x_t | x_{t - 1}) m_{t - 1}[x] +\end{equation} + +Using this method, the most likely sequence in the past $t - 1$ states, also known as the most likely explanation, will be available in $a_t[x]$. + +In our example, we can use the Mini-Viterbi algorithm starting from $x_1 = The\ weather\ n\ days\ ago$, and the most likely sequence would be available in $a_{n + 1}[x_{today}]$. + +## Hidden Markov Model +Let's start with the following scenario: + +We have two friends Alice and Bob. They live very far apart. So, they talk on the phone. Bob has a mood that changes based on the weather. Whenever they talk on the phone, Bob tells Alice his mood. + +Suppose that if the weather is sunny, then Bob is happy with a probability of 80% and grumpy with a probability of 20%. Likewise, if the weather is rainy, Bob is grumpy with a probability of 60% and happy with a probability of 40%. + +Whenever Bob tells Alice that he is happy, she may guess that the weather is sunny regarding the probabilities discussed above. Similarly, if Bob tells Alice that he is grumpy, she may suppose rainy weather. Notice that there may be some exceptions that despite the rainy weather, Bob is happy, or despite the sunny weather, he is grumpy. + +Now, we get the time involved in our problem. Suppose that if today is sunny, tomorrow will be sunny with the probability of 80%, and rainy with the probability of 20%. Likewise, if today is rainy, tomorrow will be rainy with the probability of 60%, and sunny with the probability of 40%. + +Consider the figure below to get a better understanding of the problem discussed above: +
+Bayes Net of Markov Model +
+ +Now, we formulate the problem formally: + +\begin{equation} +{E}_t: Bob's\;mood\;at\;time\;t\;(Observation) \in \{happy, grumpy\}\\ +{X}_t: Weather's\;condition\;at\;time\;t\;(State) \in \{sunny, rainy\} +\end{equation} +So, the question is, how can Alice infer the weather’s condition (state) regarding Bob’s mood (observation)? + +The problem discussed above is an example of the Hidden Markov Model (HMM). It is not possible to model the problem as a Markov Chain in many problems. So, how can we model these problems? + +Most of the time, it is possible to use HMM! If the problem has a superior state that is a Markov Process, we can make an assumption about this superior state (i.e. our belief), and update it as we observe the problem’s inputs. + +In the problem discussed before, our belief is weather’s condition that can be updated regarding the observation, Bob’s mood. + +Let’s get into HMM’s structure. We start with a simple Markov chain. At each state, we add new nodes for the inputs of the problem. These nodes are called observation nodes and act like agent sensors. Note that each observation node solely relies on its corresponding state. + +Take a closer look at the figure below to find out HMM's structure: +
+Bayes Net of Markov Model +
+
+ +Regarding the structure above, an HMM is defined by: +- $P(X_1)$ as initial distribution +- $P(X_t \mid X_{t-1})$ as transitions distribution +- $P(E_t \mid X_t)$ as emissions distribution + +We will dive into more mathematics in the next part. + +### Chain Rule and Joint Distribution of Hidden Markov Model +Let's denote $X_t,X_{t-1},\cdots,X_2,X_1$ by $X_{t:1}$. Similarly, denote $E_t,E_{t-1},\cdots,E_2,E_1$ by $E_{t:1}$. + +We are going to calculate joint distribution of HMM. + +From the Chain Rule, every joint distribution over $X_{t:1}, E_{t:1}$ can be written as: + +\begin{equation} +P(X_{t:1}, E_{t:1})=P(X_1)P(E_1 \mid X_1)P(X_2\mid X_1,E_1)P(E_2\mid X_{2:1},E_1)\cdots P(X_t\mid X_{t-1:1},E_{t-1:1})P(E_t\mid X_{t:1},E_{t-1:1}) +\end{equation} +Regarding the HMM's structure, we can assume that: + +\begin{equation} +\forall t>1:\;\;\;X_t \perp \!\!\! \perp X_{t-2:1},E_{t-1:1} \mid X_{t-1}\;\;\;and\;\;\;E_t \perp \!\!\! \perp X_{t-1:1},E_{t-1:1} \mid X_{t} +\end{equation} +Recall that every node given its parents is independent of its non-descendants. + +Using the properties above, we get: +\begin{equation} +P(X_{t:1}, E_{t:1})=P(X_1)P(E_1 \mid X_1)\prod_{i = 2}^{t} P(X_i \mid X_{i-1})P(E_i \mid X_i) +\end{equation} + +### Hidden Markov Model Applications +Hidden Markov models are known for their applications in various fields, such as: +- Signal processing +- Information theory +- Pattern recognition +- Statistical mechanics +- Physics +- Chemistry +- Economics +- Finance +- Thermodynamics + + + +Below, some of its popular applications are described in more details: + +#### Speech recognition HMMs +The main core of HMM-based speech recognition systems is the Viterbi algorithm which will be discussed in the following section. The Viterbi algorithm uses dynamic programming to find out the best alignment between the input speech and a given speech model. Nowadays, most Large Vocabulary Continuous Speech Recognition (LVCSR) systems are based on HMMs. + +In speech recognition HMMs, the continuous-valued acoustic signals are our observations. States are particular positions in specific words. + +
+speech recognition +
+ + + +#### Machine translation HMMs +A simple approach would be the mechanical substitution of words in one language for words in another. However, this won't result in coherent translations most of the time. + +Better results can be achieved using attention-based neural machine translation (NMT) models. These models focus on specific source positions to produce a translation with considerable improvements over pure encoder-decoder models. To achieve this, We use a Hidden Markov Model (HMM) consisting of neural network-based alignment and lexicon models, which are trained jointly using the forward-backward algorithm. + +In machine translation HMMs, observations are millions of words and corresponding states are translation options. + +
+machine translation +
+ +#### Robot tracking HMMs +Hidden Markov Models (HMM) are applied to interoceptive data acquired by a moving robot when contouring an environment. To track the robot accurately, the robot needs a sufficient set of sensors. + +In Robot tracking HMMs, observations are sensors' continuous perceptions and corresponding states are positions in the map. + +
+machine translation +
+ +## Conclusion +In this note, we first introduced Temporal Probability Model and discussed why we need them to model problems that vary over time. We then introduce Markov Models, explain their properties, and calculate their joint distribution with Chain Rule. + +Then, we talked about the Mini-Forward algorithm and how it can help us predict the state of the system in the future. After that, we talked about stationary distributions help us to know will eventually end up in the same distribution no matter what the initial distribution is. + +Then, we discussed a popular family of temporal models, called Hidden Markov Models for probabilistic sequence classification. In these models, we link a sequence of observations to a sequence of hidden states that explains those observations. Inference, as our goal, was the process of discovering the sequence of hidden states, given the sequence +of observations. Finally, we introduced some of the HMM's well-known applications. + +## References +* [Artificial Intelligence: A Modern Approach, 4th Edition](http://aima.cs.berkeley.edu/) +* [CE417 Lecture Slides, Prof. Rohabn, Sharif University of Technology](http://ce.sharif.edu/courses/00-01/1/ce417-1/resources/root/Slides/PDF/Session%2015_16.pdf) +* [CSC545 Lecture Slides, University of Miami](https://www.cs.miami.edu/home/visser/csc545-files/ProbabilisticReasoningOverTime.pdf) +* [Speech and Language Processing. Daniel Jurafsky & James H. Martin, Stanford University](https://web.stanford.edu/~jurafsky/slp3/A.pdf) +* [University of Tennesee Lecture Slides](https://nanopdf.com/download/chapter-15-probabilistic-reasoning-over-time_pdf) +* [Paulo E. Santos (FEI - Sao Paulo) Slides](https://fei.edu.br/~psantos/slidesIA/CAP15-DBN.pdf) +* [CS 188: Artificial Intelligence Lecture Slides, Dan Klein, Spring 2006](https://inst.eecs.berkeley.edu/~cs188/sp06/slides/cs188%20lecture%2018%20--%20HMMs%20(2pp).pdf) + +_\*Some conceptual pictures used in this note are under courtesy of [VectorStock Media](https://vectorstock.com)_ \ No newline at end of file diff --git a/notebooks/12_temporal_probability_models/metadata.yml b/notebooks/12_temporal_probability_models/metadata.yml new file mode 100644 index 00000000..544564f5 --- /dev/null +++ b/notebooks/12_temporal_probability_models/metadata.yml @@ -0,0 +1,25 @@ +title: # shown on browser tab + +header: + title: # title of your notebook + description: # short description of your notebook + +authors: + label: + position: top + content: + # list of notebook authors + - name: # name of author + role: Author # change this if you want + contact: + # list of contact information + - link: 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Date: Sat, 18 Dec 2021 00:01:49 +0330 Subject: [PATCH 04/33] Update index.md --- notebooks/12_temporal_probability_models/index.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/notebooks/12_temporal_probability_models/index.md b/notebooks/12_temporal_probability_models/index.md index a0d822a8..534c0d19 100644 --- a/notebooks/12_temporal_probability_models/index.md +++ b/notebooks/12_temporal_probability_models/index.md @@ -42,7 +42,7 @@ So far, we have learned how to deal with probabilistic reasoning in static world Let' start with an example of a static problem: >Imagine that your car has broken down. You may guess that the problem is related to the engine, batteries, or other car parts. You may take the vehicle to a car repair shop to diagnose the problem and fix it. You know that during the diagnosis process, the broken part remains broken, so the state of the car doesn't change.

-car repair +car repair
@@ -51,7 +51,7 @@ But our problems aren't always static. Consider the below example:
-diabetes +diabetes
@@ -387,4 +387,4 @@ of observations. Finally, we introduced some of the HMM's well-known application * [Paulo E. Santos (FEI - Sao Paulo) Slides](https://fei.edu.br/~psantos/slidesIA/CAP15-DBN.pdf) * [CS 188: Artificial Intelligence Lecture Slides, Dan Klein, Spring 2006](https://inst.eecs.berkeley.edu/~cs188/sp06/slides/cs188%20lecture%2018%20--%20HMMs%20(2pp).pdf) -_\*Some conceptual pictures used in this note are under courtesy of [VectorStock Media](https://vectorstock.com)_ \ No newline at end of file +_\*Some conceptual pictures used in this note are under courtesy of [VectorStock Media](https://vectorstock.com)_ From 62945d3c53841b83ca4a968833319d274fcbb349 Mon Sep 17 00:00:00 2001 From: Arshan Dalili <57400500+arshandalili@users.noreply.github.com> Date: Sat, 18 Dec 2021 00:11:55 +0330 Subject: [PATCH 05/33] Update index.md --- notebooks/12_temporal_probability_models/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/12_temporal_probability_models/index.md b/notebooks/12_temporal_probability_models/index.md index 534c0d19..7aa56f35 100644 --- a/notebooks/12_temporal_probability_models/index.md +++ b/notebooks/12_temporal_probability_models/index.md @@ -269,7 +269,7 @@ Now, we get the time involved in our problem. Suppose that if today is sunny, to Consider the figure below to get a better understanding of the problem discussed above:
-Bayes Net of Markov Model +Bayes Net of Markov Model
Now, we formulate the problem formally: From 7715516de516c64d12833d0f5f6a6cef3fbe6759 Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Sat, 18 Dec 2021 00:30:55 +0330 Subject: [PATCH 06/33] Edit files --- .../12_temporal_probability_models/index.md | 20 ++----------------- .../metadata.yml | 2 +- 2 files changed, 3 insertions(+), 19 deletions(-) diff --git a/notebooks/12_temporal_probability_models/index.md b/notebooks/12_temporal_probability_models/index.md index 8df4b0b8..6e15a63d 100644 --- a/notebooks/12_temporal_probability_models/index.md +++ b/notebooks/12_temporal_probability_models/index.md @@ -42,11 +42,7 @@ So far, we have learned how to deal with probabilistic reasoning in static world Let' start with an example of a static problem: >Imagine that your car has broken down. You may guess that the problem is related to the engine, batteries, or other car parts. You may take the vehicle to a car repair shop to diagnose the problem and fix it. You know that during the diagnosis process, the broken part remains broken, so the state of the car doesn't change.
-<<<<<<< HEAD -car repair -======= car repair ->>>>>>> 62945d3c53841b83ca4a968833319d274fcbb349
@@ -55,11 +51,7 @@ But our problems aren't always static. Consider the below example:
-<<<<<<< HEAD -diabetes -======= diabetes ->>>>>>> 62945d3c53841b83ca4a968833319d274fcbb349
@@ -75,7 +67,7 @@ Retake variables of the diabetes example. We know that these variables can chang >_**Hence we can model the diabetes status of a patient using a Temporal Probability Model.**_ -There are some variables like measured blood sugar and pulse rate that are measurable, and we denote them as at time $t$. +There are some variables like measured blood sugar and pulse rate that are measurable, and we denote them as ${E}_t$ at time $t$. On the other hand, we have some unmeasurable variables like stomach content and blood sugar. (Notice that measured blood sugar and blood sugar are considered different variables since measured blood sugar has some errors.) and at time $t$, we use ${X}_t$ to denote them. In a nutshell, we have: @@ -277,11 +269,7 @@ Now, we get the time involved in our problem. Suppose that if today is sunny, to Consider the figure below to get a better understanding of the problem discussed above:
-<<<<<<< HEAD -Bayes Net of Markov Model -======= Bayes Net of Markov Model ->>>>>>> 62945d3c53841b83ca4a968833319d274fcbb349
Now, we formulate the problem formally: @@ -399,8 +387,4 @@ of observations. Finally, we introduced some of the HMM's well-known application * [Paulo E. Santos (FEI - Sao Paulo) Slides](https://fei.edu.br/~psantos/slidesIA/CAP15-DBN.pdf) * [CS 188: Artificial Intelligence Lecture Slides, Dan Klein, Spring 2006](https://inst.eecs.berkeley.edu/~cs188/sp06/slides/cs188%20lecture%2018%20--%20HMMs%20(2pp).pdf) -<<<<<<< HEAD -_\*Some conceptual pictures used in this note are under courtesy of [VectorStock Media](https://vectorstock.com)_ -======= -_\*Some conceptual pictures used in this note are under courtesy of [VectorStock Media](https://vectorstock.com)_ ->>>>>>> 62945d3c53841b83ca4a968833319d274fcbb349 +_\*Some conceptual pictures used in this note are under courtesy of [VectorStock Media](https://vectorstock.com)_ \ No newline at end of file diff --git a/notebooks/12_temporal_probability_models/metadata.yml b/notebooks/12_temporal_probability_models/metadata.yml index 952fc538..555e8568 100644 --- a/notebooks/12_temporal_probability_models/metadata.yml +++ b/notebooks/12_temporal_probability_models/metadata.yml @@ -17,7 +17,7 @@ authors: icon: fab fa-github - link: https://github.com/SJahanzad icon: fab fa-github - - link: https://github.com/https://github.com/MohammadHejri + - link: https://github.com/MohammadHejri icon: fab fa-github # optionally add other contact information like # - link: # contact link From 900a6af5a162106909c556a68b29e9f42b5e1e9f Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Sat, 18 Dec 2021 00:39:59 +0330 Subject: [PATCH 07/33] Test new LaTeX --- .../12_temporal_probability_models/index.md | 22 +++++++++---------- 1 file changed, 11 insertions(+), 11 deletions(-) diff --git a/notebooks/12_temporal_probability_models/index.md b/notebooks/12_temporal_probability_models/index.md index 6e15a63d..d1535dc2 100644 --- a/notebooks/12_temporal_probability_models/index.md +++ b/notebooks/12_temporal_probability_models/index.md @@ -216,11 +216,11 @@ Finally, let's take a look at the state of weather with initial observation of r An interesting question to ask in problems modeled using Markov chains is about the state as we continue simulating for a long time. As we simulate further into the future, the Moarkov chain becomes longer and uncertainty is accumulated. -In the general case, we cannot say what the state will be in the far future. However, for most Markov chains, we will eventually end up in the same distribution no matter what the initial distribution is. In other words, the distribution we end up in is independent of the initial distribution. In such cases, the distribution we end up with is called the **stationary distribution** of the chain. This distribution is denoted by $P_\infty$ and satisfies the condition below. +In the general case, we cannot say what the state will be in the far future. However, for most Markov chains, we will eventually end up in the same distribution no matter what the initial distribution is. In other words, the distribution we end up in is independent of the initial distribution. In such cases, the distribution we end up with is called the **stationary distribution** of the chain. This distribution is denoted by and satisfies the condition below. -\begin{equation} - P_{\infty}(X) = P_{\infty + 1}(X) = \sum_{x} P(X|x) P_{\infty}(x) -\end{equation} +
+ +
In the weather example we mentioned earlier, the stationary distribution denotes the probability that each day will be sunny or rainy in general. @@ -237,12 +237,13 @@ Again, a slow answer would be to enumerate all possible combinations and calcula In the Mini-Viterbi algorithm, we define $m_t[x]$ and $a_t[x]$ as: -\begin{equation} - m_t[x] = \max_{x_{1:t - 1}} P(x_{1:t - 1}, x) -\end{equation} -\begin{equation} - a_t[x] = arg\!\max_{x_{1:t - 1}} P(x_{1:t - 1}, x) -\end{equation} +
+ +
+
+
+ +
The equation above can be rewritten to: \begin{equation} @@ -273,7 +274,6 @@ Consider the figure below to get a better understanding of the problem discussed Now, we formulate the problem formally: - \begin{equation} {E}_t: Bob's\;mood\;at\;time\;t\;(Observation) \in \{happy, grumpy\}\\ {X}_t: Weather's\;condition\;at\;time\;t\;(State) \in \{sunny, rainy\} From 037003c249195ebf395f2704dd9536854a39fb5e Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Sat, 18 Dec 2021 01:01:51 +0330 Subject: [PATCH 08/33] Change LaTeX to image --- .../doc/index.md | 390 ++++++++++++++++++ .../doc/teximg/tex_img_0_39OCG.png | Bin 0 -> 245 bytes .../doc/teximg/tex_img_10_AYRJS.png | Bin 0 -> 277 bytes .../doc/teximg/tex_img_11_BRTR9.png | Bin 0 -> 237 bytes .../doc/teximg/tex_img_12_80OWW.png | Bin 0 -> 252 bytes .../doc/teximg/tex_img_13_0CF2J.png | Bin 0 -> 181 bytes .../doc/teximg/tex_img_14_9E376.png | Bin 0 -> 213 bytes .../doc/teximg/tex_img_15_P34IE.png | Bin 0 -> 181 bytes 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Markov Models](#Joint-Distributions-of-Markov-Models) +* [Chain Rule](#Chain-Rule) + +[Mini-Forward Algorithm](#Mini-Forward-Algorithm) + +[Stationary Distribution](#Stationary-Distribution) + +[Hidden Markov Model](#Hidden-Markov-Model) +* [Chain Rule and Joint Distribution of Hidden Markov Model](#Chain-Rule-and-Joint-Distribution-of-Hidden-Markov-Model) +* [Hidden Markov Models Applications](#Hidden-Markov-Model-Applications) + +[Conclusion](#Conclusion) + +[References](#References) + + + + + +## Introduction +In this lecture note, we are going to talk about **_Temporal Probability Models_**. We will discuss what they are and how they can help us model problems that are subject to change over time. + + +## Reasoning over time or space +So far, we have learned how to deal with probabilistic reasoning in static worlds. That means each random variable has a fixed value and doesn't change over time. However, most of our problems don't have this static behavior, and the state varies over time (or space). + +Let' start with an example of a static problem: +>Imagine that your car has broken down. You may guess that the problem is related to the engine, batteries, or other car parts. You may take the vehicle to a car repair shop to diagnose the problem and fix it. You know that during the diagnosis process, the broken part remains broken, so the state of the car doesn't change. +
+car repair +
+ + +But our problems aren't always static. Consider the below example: +>Take diabetes management of a patient as our problem. We know that, for instance, the patient's blood sugar isn't constant and will vary over time, so there is a dynamic in the variables used to describe the patient's state (such as the blood sugar and insulin). + + +
+diabetes +
+ + +As you can see, the patient's status varies over time, and our approaches in static probabilistic reasoning won't work for these problems. So we should look for other approaches to solve them. + +Temporal Probability Models can make it easier for us to model and solve these types of problems. + + +### What is a Temporal Probability Model? + +As the diabetes example above shows, we need to introduce **time** (or space) for our models. +Retake variables of the diabetes example. We know that these variables can change over time, so to manage the diabetes of the patient, we slice time at specific points and keep track of this evidence to evaluate the current state and predict the future diabetic condition of the patient. + +>_**Hence we can model the diabetes status of a patient using a Temporal Probability Model.**_ + +There are some variables like measured blood sugar and pulse rate that are measurable, and we denote them as ${E}_t$ at time $t$. + +On the other hand, we have some unmeasurable variables like stomach content and blood sugar. (Notice that measured blood sugar and blood sugar are considered different variables since measured blood sugar has some errors.) and at time $t$, we use ${X}_t$ to denote them. +In a nutshell, we have: + +\begin{equation} +{E}_t = Observable\;Variable\;at\;time\;t\\ +{X}_t = NonObservable\;Variable\;at\;time\;t +\end{equation} + + + +### Markov Models +Let's define a random variable $X$ and call its value at a given time $t$ the state of $X$ and denote it as $X_t$. If in our model the future state of $X$ depends only on the current state of $X$ and not on states that had occurred before, we call it a **_First-order Markov Model_**, or simply a **_Markov Model_**. Bayesian network of this model looks like the below network and is referred to as a **_Markov Chain_**: +
+Bayes Net of Markov Model +
+ +In a Markov chain, we can write the Morkov assumption as: + +\begin{equation} +P(X_n=x_n|X_{1:n-1}=x_{1:n-1}) = P(X_n = x_n|X_{n - 1}=x_{n - 1}) +\end{equation} + +*Note:* We also define $X_{a:b}$ to denote a set of variables from $X_a$ to $X_b$. + +\begin{equation} +X_{a:b} = X_a, X_{a+1}, \cdots, X_{b-1}, X_b +\end{equation} + + +In a similar way, we can use **_$i$th-order Markov Models_** to model situations in which the future state of $X$ depends only on $i$ recent states and not on states that had occrred before. In this case we have: + +\begin{equation} +P(X_n=x_n|X_{1:n-1}=x_{1:n-1}) = P(X_n = x|X_{n - i:n - 1}=x_{n - i : n - 1}) +\end{equation} + +### Joint Distributions of Markov Models + +Take $X$ as a Markov Model and consider its Bayesian network (depicted below). +
+Bayes Net of Markov Model +
+ +We can quickly notice that for every $1 \lt i \leq n$ , if we are given $X_{i-1}$, then **$X_i$ is independent of $X_1, X_2, \cdots, X_{i-2}$** since there is only one inactive path between each of them and $X_i$. (Regarding the fact that $X_{i-1}$ is given) +So we have: + +\begin{equation} + X_i \perp \!\!\! \perp X_1, X_2, \cdots, X_{i-2} \vert X_{i-1} +\end{equation} + +We define two parameters here: +* **Transition Probabilities**: + + \begin{equation} + P(X_t \vert X_{t-1}) + \end{equation} + + It shows how each state is related to its previous state and how states develop. + The **_Stationary assumption_** in Markov models ensures that transition probabilities don't change over time. +* **Initial State Probability**: + + \begin{equation} + P(X_1) + \end{equation} + + It is the probability of the initial state ($X_1$), and we usually know its value. + + We can find the joint distribution of X regarding the Bayesian network and the parameters we defined. If we consider $X_1$ to $X_n$, then we have: + + \begin{equation} + P(X_1, X_2, \cdots, X_{n-1}, X_n) = P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n-1}\vert X_{n-2})P(X_{n}\vert X_{n-1})\\ \Rightarrow P(X_1, X_2, \cdots, X_{n-1}, X_n) =P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) + \end{equation} + +In the next section, we will prove this joint distribution using Chain Rule and independence. + + + + +### Chain Rule +Take $X$ as a Markov Model and consider its Bayesian network (depicted below). +
+Bayes Net of Markov Model +
+ +We know that from Chain Rule, every joint distribution of $X_1$ to $X_n$ can be written as: + + \begin{equation} + P(X_1, X_2, \cdots, X_{n-1}, X_n) = P(X_1)P(X_2\vert X_1)P(X_3\vert X_1, X_2)\cdots P(X_{n}\vert X_1,X_2,\cdots X_{n-1}) + \end{equation} + +Now, due to the independence we saw earlier, we can assume: + +\begin{equation} + X_3 \perp \!\!\! \perp X_1\vert X_2, X_4\perp \!\!\! \perp X_1,X_2\vert X_3, \cdots, X_{n}\perp \!\!\! \perp X_1,X_2\cdots X_{n-2}\vert X_{n-1} +\end{equation} + +So, after applying independence to Chain Rule, we have: + + \begin{equation} + P(X_1, X_2, \cdots, X_{n-1}, X_n)= P(X_1)P(X_2\vert X_1)P(X_3\vert X_1, X_2)\cdots P(X_{n}\vert X_1,X_2,\cdots X_{n-1})\\=P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n}\vert X_{n-1})\\=P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) + \end{equation} + +which is the same result we get from the previous section. + +## Mini-Forward Algorithm + +Imagine you want to go on a trip with your friends. Your friends think that trips in sunny days are more fun, so they ask you to schedule the trip for a sunny day. You know that you cannot be completely certain whether the $n$th day after today is going to be sunny or rainy, so what will you do? + +We can solve this problem easily if we assume a Markov model. Suppose that we have measured the transition probabilities for this model and know that if some day is sunny, the next day will be sunny with probability $P(X_{n + 1} = sunny | X_{n} = sunny) = 0.9$ and will be rainy otherwise. Similarly, if some day is rainy, the next day will be rainy with probability $P(X_{n + 1} = rainy | X_{n} = rainy) = 0.7$ and will be sunny otherwise. We also know the initial probability because we know today's weather. + +Now that we have a good model, we need an efficient approach to find the probability of each outcome in the $n$th day after today. A naive approach would be to enumerate all possible sequences of length $n$ which end in each outcome and then add up their probabilities. This will do the work, but it will require a lot of time and effort and is too slow. A better approach is to store our incremental *belief* about the weather and to update it as we consider proceed twords the $n$th day in our calculations. This approach is called the **Mini-Forward Algorithm**. + +In the Mini-Forward algorithm, we use the probabilities for the initial state and forward simulation to calculate the probabilities in the future states. The probability of outcome $x_i$ in the state $X_n$ using this algorithm is: + +\begin{equation} + P(X_n = x_i) = \sum_{x_j} P(X_n = x_i | X_{n - 1} = x_j) P(X_{n - 1} = x_j) +\end{equation} + +This can be easily computed for each state because we know the initial probability $P(X_1 = x_k)$ for every $k$ and the transition probabilities for each state can also be calculated using this algorothm. + +Now that we have this solution, we can say that the probability that the $n$th day after today will be sunny in our example is: + +\begin{equation} + P(X_{n + 1} = sunny) = \sum_{x_j \in \{sunny, rainy\}} P(X_n = sunny | X_{n - 1} = x_j) P(X_{n - 1} = x_j) +\end{equation} + +\begin{equation} + \Rightarrow P(X_{n + 1} = sunny) = 0.9 \times P(X_n = sunny | X_{n - 1} = sunny) + 0.3 \times P(X_n = sunny | X_{n - 1} = rainy) +\end{equation} + +Now, let's take a look at the state of weather with initial observation of sun: +
+Initial Sunny Day +
+
+ +Finally, let's take a look at the state of weather with initial observation of rain: + +
+Initial Sunny Day +
+ + + +## Stationary Distribution + +An interesting question to ask in problems modeled using Markov chains is about the state as we continue simulating for a long time. As we simulate further into the future, the Moarkov chain becomes longer and uncertainty is accumulated. + +In the general case, we cannot say what the state will be in the far future. However, for most Markov chains, we will eventually end up in the same distribution no matter what the initial distribution is. In other words, the distribution we end up in is independent of the initial distribution. In such cases, the distribution we end up with is called the **stationary distribution** of the chain. This distribution is denoted by $P_\infty$ and satisfies the condition below. + +\begin{equation} + P_{\infty}(X) = P_{\infty + 1}(X) = \sum_{x} P(X|x) P_{\infty}(x) +\end{equation} + +In the weather example we mentioned earlier, the stationary distribution denotes the probability that each day will be sunny or rainy in general. + + +## Most Likely Explanation: Mini-Viterbi Algorithm + +Sometimes we would like to find out what happened in the past using the knowledge that we have in present. Consider the trip we discussed earlier as an example. Imagine that you and your friends want to go outside, but some of your friends warn you that everything might be wet if it had rained in the past $n$ days. Since your destination is too far, you do not know about its past, so we need to reason about the most likely sequence of your destination's weather in the past $n$ days. The course of this problem can be shown using the following figure. + +
+Weather CPT +
+ +Again, a slow answer would be to enumerate all possible combinations and calculate the probabilities. However, we can calculate the most likely sequence inductively, remember the incremental probabilities, and update the most likely sequence as we proceed. This approach is called the **Mini-Viterbi Algorithm**. + +In the Mini-Viterbi algorithm, we define $m_t[x]$ and $a_t[x]$ as: + +\begin{equation} + m_t[x] = \max_{x_{1:t - 1}} P(x_{1:t - 1}, x) +\end{equation} +\begin{equation} + a_t[x] = arg\!\max_{x_{1:t - 1}} P(x_{1:t - 1}, x) +\end{equation} + +The equation above can be rewritten to: +\begin{equation} + m_1[x] = P(X_1 = x1) +\end{equation} +\begin{equation} + m_t[x] = \max_{x_{1:t - 1}} P(x_t | x_{t - 1}) m_{t - 1}[x] +\end{equation} + +Using this method, the most likely sequence in the past $t - 1$ states, also known as the most likely explanation, will be available in $a_t[x]$. + +In our example, we can use the Mini-Viterbi algorithm starting from $x_1 = The\ weather\ n\ days\ ago$, and the most likely sequence would be available in $a_{n + 1}[x_{today}]$. + +## Hidden Markov Model +Let's start with the following scenario: + +We have two friends Alice and Bob. They live very far apart. So, they talk on the phone. Bob has a mood that changes based on the weather. Whenever they talk on the phone, Bob tells Alice his mood. + +Suppose that if the weather is sunny, then Bob is happy with a probability of 80% and grumpy with a probability of 20%. Likewise, if the weather is rainy, Bob is grumpy with a probability of 60% and happy with a probability of 40%. + +Whenever Bob tells Alice that he is happy, she may guess that the weather is sunny regarding the probabilities discussed above. Similarly, if Bob tells Alice that he is grumpy, she may suppose rainy weather. Notice that there may be some exceptions that despite the rainy weather, Bob is happy, or despite the sunny weather, he is grumpy. + +Now, we get the time involved in our problem. Suppose that if today is sunny, tomorrow will be sunny with the probability of 80%, and rainy with the probability of 20%. Likewise, if today is rainy, tomorrow will be rainy with the probability of 60%, and sunny with the probability of 40%. + +Consider the figure below to get a better understanding of the problem discussed above: +
+Bayes Net of Markov Model +
+ +Now, we formulate the problem formally: + +\begin{equation} +{E}_t: Bob's\;mood\;at\;time\;t\;(Observation) \in \{happy, grumpy\}\\ +{X}_t: Weather's\;condition\;at\;time\;t\;(State) \in \{sunny, rainy\} +\end{equation} +So, the question is, how can Alice infer the weather’s condition (state) regarding Bob’s mood (observation)? + +The problem discussed above is an example of the Hidden Markov Model (HMM). It is not possible to model the problem as a Markov Chain in many problems. So, how can we model these problems? + +Most of the time, it is possible to use HMM! If the problem has a superior state that is a Markov Process, we can make an assumption about this superior state (i.e. our belief), and update it as we observe the problem’s inputs. + +In the problem discussed before, our belief is weather’s condition that can be updated regarding the observation, Bob’s mood. + +Let’s get into HMM’s structure. We start with a simple Markov chain. At each state, we add new nodes for the inputs of the problem. These nodes are called observation nodes and act like agent sensors. Note that each observation node solely relies on its corresponding state. + +Take a closer look at the figure below to find out HMM's structure: +
+Bayes Net of Markov Model +
+
+ +Regarding the structure above, an HMM is defined by: +- $P(X_1)$ as initial distribution +- $P(X_t \mid X_{t-1})$ as transitions distribution +- $P(E_t \mid X_t)$ as emissions distribution + +We will dive into more mathematics in the next part. + +### Chain Rule and Joint Distribution of Hidden Markov Model +Let's denote $X_t,X_{t-1},\cdots,X_2,X_1$ by $X_{t:1}$. Similarly, denote $E_t,E_{t-1},\cdots,E_2,E_1$ by $E_{t:1}$. + +We are going to calculate joint distribution of HMM. + +From the Chain Rule, every joint distribution over $X_{t:1}, E_{t:1}$ can be written as: + +\begin{equation} +P(X_{t:1}, E_{t:1})=P(X_1)P(E_1 \mid X_1)P(X_2\mid X_1,E_1)P(E_2\mid X_{2:1},E_1)\cdots P(X_t\mid X_{t-1:1},E_{t-1:1})P(E_t\mid X_{t:1},E_{t-1:1}) +\end{equation} +Regarding the HMM's structure, we can assume that: + +\begin{equation} +\forall t>1:\;\;\;X_t \perp \!\!\! \perp X_{t-2:1},E_{t-1:1} \mid X_{t-1}\;\;\;and\;\;\;E_t \perp \!\!\! \perp X_{t-1:1},E_{t-1:1} \mid X_{t} +\end{equation} +Recall that every node given its parents is independent of its non-descendants. + +Using the properties above, we get: +\begin{equation} +P(X_{t:1}, E_{t:1})=P(X_1)P(E_1 \mid X_1)\prod_{i = 2}^{t} P(X_i \mid X_{i-1})P(E_i \mid X_i) +\end{equation} + +### Hidden Markov Model Applications +Hidden Markov models are known for their applications in various fields, such as: +- Signal processing +- Information theory +- Pattern recognition +- Statistical mechanics +- Physics +- Chemistry +- Economics +- Finance +- Thermodynamics + + + +Below, some of its popular applications are described in more details: + +#### Speech recognition HMMs +The main core of HMM-based speech recognition systems is the Viterbi algorithm which will be discussed in the following section. The Viterbi algorithm uses dynamic programming to find out the best alignment between the input speech and a given speech model. Nowadays, most Large Vocabulary Continuous Speech Recognition (LVCSR) systems are based on HMMs. + +In speech recognition HMMs, the continuous-valued acoustic signals are our observations. States are particular positions in specific words. + +
+speech recognition +
+ + + +#### Machine translation HMMs +A simple approach would be the mechanical substitution of words in one language for words in another. However, this won't result in coherent translations most of the time. + +Better results can be achieved using attention-based neural machine translation (NMT) models. These models focus on specific source positions to produce a translation with considerable improvements over pure encoder-decoder models. To achieve this, We use a Hidden Markov Model (HMM) consisting of neural network-based alignment and lexicon models, which are trained jointly using the forward-backward algorithm. + +In machine translation HMMs, observations are millions of words and corresponding states are translation options. + +
+machine translation +
+ +#### Robot tracking HMMs +Hidden Markov Models (HMM) are applied to interoceptive data acquired by a moving robot when contouring an environment. To track the robot accurately, the robot needs a sufficient set of sensors. + +In Robot tracking HMMs, observations are sensors' continuous perceptions and corresponding states are positions in the map. + +
+machine translation +
+ +## Conclusion +In this note, we first introduced Temporal Probability Model and discussed why we need them to model problems that vary over time. We then introduce Markov Models, explain their properties, and calculate their joint distribution with Chain Rule. + +Then, we talked about the Mini-Forward algorithm and how it can help us predict the state of the system in the future. After that, we talked about stationary distributions help us to know will eventually end up in the same distribution no matter what the initial distribution is. + +Then, we discussed a popular family of temporal models, called Hidden Markov Models for probabilistic sequence classification. In these models, we link a sequence of observations to a sequence of hidden states that explains those observations. Inference, as our goal, was the process of discovering the sequence of hidden states, given the sequence +of observations. Finally, we introduced some of the HMM's well-known applications. + +## References +* [Artificial Intelligence: A Modern Approach, 4th Edition](http://aima.cs.berkeley.edu/) +* [CE417 Lecture Slides, Prof. Rohabn, Sharif University of Technology](http://ce.sharif.edu/courses/00-01/1/ce417-1/resources/root/Slides/PDF/Session%2015_16.pdf) +* [CSC545 Lecture Slides, University of Miami](https://www.cs.miami.edu/home/visser/csc545-files/ProbabilisticReasoningOverTime.pdf) +* [Speech and Language Processing. Daniel Jurafsky & James H. Martin, Stanford University](https://web.stanford.edu/~jurafsky/slp3/A.pdf) +* [University of Tennesee Lecture Slides](https://nanopdf.com/download/chapter-15-probabilistic-reasoning-over-time_pdf) +* [Paulo E. Santos (FEI - Sao Paulo) Slides](https://fei.edu.br/~psantos/slidesIA/CAP15-DBN.pdf) +* [CS 188: Artificial Intelligence Lecture Slides, Dan Klein, Spring 2006](https://inst.eecs.berkeley.edu/~cs188/sp06/slides/cs188%20lecture%2018%20--%20HMMs%20(2pp).pdf) + +_\*Some conceptual pictures used in this note are under courtesy of [VectorStock Media](https://vectorstock.com)_ \ No newline at end of file diff --git a/notebooks/12_temporal_probability_models/doc/teximg/tex_img_0_39OCG.png b/notebooks/12_temporal_probability_models/doc/teximg/tex_img_0_39OCG.png new file mode 100644 index 0000000000000000000000000000000000000000..3130ae498c33cc18005d1bf007d978fd1535a7ed GIT binary patch literal 245 zcmeAS@N?(olHy`uVBq!ia0vp^f55H;$Qn~>?A+G=b{|7R6?b;O*5+WcV z;Ns#^R#ujlmUidPosN!<88c?Qd-v|lnKMdCN~>0_dhPN$1*n>{z$3C4XxI@DW}K?c zCk+(L^K@|x;gC&ENHF5`*49j5U_8pkwv5Ffn`=T0LmI1bL4kHde9Oo1{b~B+|Cuy>;Ngh6#<0CmZ-PPMRG*^TmO+ lYyk_)hj)ypniy|evoY*<64vRiI|uR@gQu&X%Q~loCIGYqRDA#d literal 0 HcmV?d00001 diff --git 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d1535dc2..75c975ab 100644 --- a/notebooks/12_temporal_probability_models/index.md +++ b/notebooks/12_temporal_probability_models/index.md @@ -42,7 +42,7 @@ So far, we have learned how to deal with probabilistic reasoning in static world Let' start with an example of a static problem: >Imagine that your car has broken down. You may guess that the problem is related to the engine, batteries, or other car parts. You may take the vehicle to a car repair shop to diagnose the problem and fix it. You know that during the diagnosis process, the broken part remains broken, so the state of the car doesn't change.
-car repair +car repair
@@ -51,7 +51,7 @@ But our problems aren't always static. Consider the below example:
-diabetes +diabetes
@@ -67,9 +67,9 @@ Retake variables of the diabetes example. We know that these variables can chang >_**Hence we can model the diabetes status of a patient using a Temporal Probability Model.**_ -There are some variables like measured blood sugar and pulse rate that are measurable, and we denote them as ${E}_t$ at time $t$. +There are some variables like measured blood sugar and pulse rate that are measurable, and we denote them as ![{E}_t](doc/teximg/tex_img_0_39OCG.png) at time ![t](doc/teximg/tex_img_1_YYV34.png). -On the other hand, we have some unmeasurable variables like stomach content and blood sugar. (Notice that measured blood sugar and blood sugar are considered different variables since measured blood sugar has some errors.) and at time $t$, we use ${X}_t$ to denote them. +On the other hand, we have some unmeasurable variables like stomach content and blood sugar. (Notice that measured blood sugar and blood sugar are considered different variables since measured blood sugar has some errors.) and at time ![t](doc/teximg/tex_img_2_GQ4EP.png), we use ![{X}_t](doc/teximg/tex_img_3_VET8W.png) to denote them. In a nutshell, we have: \begin{equation} @@ -80,7 +80,7 @@ In a nutshell, we have: ### Markov Models -Let's define a random variable $X$ and call its value at a given time $t$ the state of $X$ and denote it as $X_t$. If in our model the future state of $X$ depends only on the current state of $X$ and not on states that had occurred before, we call it a **_First-order Markov Model_**, or simply a **_Markov Model_**. Bayesian network of this model looks like the below network and is referred to as a **_Markov Chain_**: +Let's define a random variable ![X](doc/teximg/tex_img_4_8A5SX.png) and call its value at a given time ![t](doc/teximg/tex_img_5_S26DU.png) the state of ![X](doc/teximg/tex_img_6_FQ2GJ.png) and denote it as ![X_t](doc/teximg/tex_img_7_79OU6.png). If in our model the future state of ![X](doc/teximg/tex_img_8_MYNF4.png) depends only on the current state of ![X](doc/teximg/tex_img_9_X5BQ6.png) and not on states that had occurred before, we call it a **_First-order Markov Model_**, or simply a **_Markov Model_**. Bayesian network of this model looks like the below network and is referred to as a **_Markov Chain_**:
Bayes Net of Markov Model
@@ -91,14 +91,14 @@ In a Markov chain, we can write the Morkov assumption as: P(X_n=x_n|X_{1:n-1}=x_{1:n-1}) = P(X_n = x_n|X_{n - 1}=x_{n - 1}) \end{equation} -*Note:* We also define $X_{a:b}$ to denote a set of variables from $X_a$ to $X_b$. +*Note:* We also define ![X_{a:b}](doc/teximg/tex_img_10_AYRJS.png) to denote a set of variables from ![X_a](doc/teximg/tex_img_11_BRTR9.png) to ![X_b](doc/teximg/tex_img_12_80OWW.png). \begin{equation} X_{a:b} = X_a, X_{a+1}, \cdots, X_{b-1}, X_b \end{equation} -In a similar way, we can use **_$i$th-order Markov Models_** to model situations in which the future state of $X$ depends only on $i$ recent states and not on states that had occrred before. In this case we have: +In a similar way, we can use **_![i](doc/teximg/tex_img_13_0CF2J.png)th-order Markov Models_** to model situations in which the future state of ![X](doc/teximg/tex_img_14_9E376.png) depends only on ![i](doc/teximg/tex_img_15_P34IE.png) recent states and not on states that had occrred before. In this case we have: \begin{equation} P(X_n=x_n|X_{1:n-1}=x_{1:n-1}) = P(X_n = x|X_{n - i:n - 1}=x_{n - i : n - 1}) @@ -106,12 +106,12 @@ P(X_n=x_n|X_{1:n-1}=x_{1:n-1}) = P(X_n = x|X_{n - i:n - 1}=x_{n - i : n - 1}) ### Joint Distributions of Markov Models -Take $X$ as a Markov Model and consider its Bayesian network (depicted below). +Take ![X](doc/teximg/tex_img_16_XZRN9.png) as a Markov Model and consider its Bayesian network (depicted below).
Bayes Net of Markov Model
-We can quickly notice that for every $1\lt i\le n$ , if we are given $X_{i-1}$, then **$X_i$ is independent of $X_1, X_2, \cdots, X_{i-2}$** since there is only one inactive path between each of them and $X_i$. (Regarding the fact that $X_{i-1}$ is given) +We can quickly notice that for every ![1 \lt i \leq n](doc/teximg/tex_img_17_HALC6.png) , if we are given ![X_{i-1}](doc/teximg/tex_img_18_Q8NV3.png), then **![X_i](doc/teximg/tex_img_19_5IG3C.png) is independent of ![X_1, X_2, \cdots, X_{i-2}](doc/teximg/tex_img_20_D73OR.png)** since there is only one inactive path between each of them and ![X_i](doc/teximg/tex_img_21_FOUL8.png). (Regarding the fact that ![X_{i-1}](doc/teximg/tex_img_22_ABEBF.png) is given) So we have: \begin{equation} @@ -133,9 +133,9 @@ We define two parameters here: P(X_1) \end{equation} - It is the probability of the initial state ($X_1$), and we usually know its value. + It is the probability of the initial state (![X_1](doc/teximg/tex_img_23_13HCZ.png)), and we usually know its value. - We can find the joint distribution of X regarding the Bayesian network and the parameters we defined. If we consider $X_1$ to $X_n$, then we have: + We can find the joint distribution of X regarding the Bayesian network and the parameters we defined. If we consider ![X_1](doc/teximg/tex_img_24_FG02S.png) to ![X_n](doc/teximg/tex_img_25_LCZQB.png), then we have: \begin{equation} P(X_1, X_2, \cdots, X_{n-1}, X_n) = P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n-1}\vert X_{n-2})P(X_{n}\vert X_{n-1})\\ \Rightarrow P(X_1, X_2, \cdots, X_{n-1}, X_n) =P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) @@ -147,12 +147,12 @@ In the next section, we will prove this joint distribution using Chain Rule and ### Chain Rule -Take $X$ as a Markov Model and consider its Bayesian network (depicted below). +Take ![X](doc/teximg/tex_img_26_16ZCN.png) as a Markov Model and consider its Bayesian network (depicted below).
Bayes Net of Markov Model
-We know that from Chain Rule, every joint distribution of $X_1$ to $X_n$ can be written as: +We know that from Chain Rule, every joint distribution of ![X_1](doc/teximg/tex_img_27_P67XU.png) to ![X_n](doc/teximg/tex_img_28_UIG2L.png) can be written as: \begin{equation} P(X_1, X_2, \cdots, X_{n-1}, X_n) = P(X_1)P(X_2\vert X_1)P(X_3\vert X_1, X_2)\cdots P(X_{n}\vert X_1,X_2,\cdots X_{n-1}) @@ -174,21 +174,21 @@ which is the same result we get from the previous section. ## Mini-Forward Algorithm -Imagine you want to go on a trip with your friends. Your friends think that trips in sunny days are more fun, so they ask you to schedule the trip for a sunny day. You know that you cannot be completely certain whether the $n$th day after today is going to be sunny or rainy, so what will you do? +Imagine you want to go on a trip with your friends. Your friends think that trips in sunny days are more fun, so they ask you to schedule the trip for a sunny day. You know that you cannot be completely certain whether the ![n](doc/teximg/tex_img_29_Z8H25.png)th day after today is going to be sunny or rainy, so what will you do? -We can solve this problem easily if we assume a Markov model. Suppose that we have measured the transition probabilities for this model and know that if some day is sunny, the next day will be sunny with probability $P(X_{n + 1} = sunny | X_{n} = sunny) = 0.9$ and will be rainy otherwise. Similarly, if some day is rainy, the next day will be rainy with probability $P(X_{n + 1} = rainy | X_{n} = rainy) = 0.7$ and will be sunny otherwise. We also know the initial probability because we know today's weather. +We can solve this problem easily if we assume a Markov model. Suppose that we have measured the transition probabilities for this model and know that if some day is sunny, the next day will be sunny with probability ![P(X_{n + 1} = sunny | X_{n} = sunny) = 0.9](doc/teximg/tex_img_30_VIR8P.png) and will be rainy otherwise. Similarly, if some day is rainy, the next day will be rainy with probability ![P(X_{n + 1} = rainy | X_{n} = rainy) = 0.7](doc/teximg/tex_img_31_GOEIM.png) and will be sunny otherwise. We also know the initial probability because we know today's weather. -Now that we have a good model, we need an efficient approach to find the probability of each outcome in the $n$th day after today. A naive approach would be to enumerate all possible sequences of length $n$ which end in each outcome and then add up their probabilities. This will do the work, but it will require a lot of time and effort and is too slow. A better approach is to store our incremental *belief* about the weather and to update it as we consider proceed twords the $n$th day in our calculations. This approach is called the **Mini-Forward Algorithm**. +Now that we have a good model, we need an efficient approach to find the probability of each outcome in the ![n](doc/teximg/tex_img_32_HP3H9.png)th day after today. A naive approach would be to enumerate all possible sequences of length ![n](doc/teximg/tex_img_33_GATCE.png) which end in each outcome and then add up their probabilities. This will do the work, but it will require a lot of time and effort and is too slow. A better approach is to store our incremental *belief* about the weather and to update it as we consider proceed twords the ![n](doc/teximg/tex_img_34_8ZF17.png)th day in our calculations. This approach is called the **Mini-Forward Algorithm**. -In the Mini-Forward algorithm, we use the probabilities for the initial state and forward simulation to calculate the probabilities in the future states. The probability of outcome $x_i$ in the state $X_n$ using this algorithm is: +In the Mini-Forward algorithm, we use the probabilities for the initial state and forward simulation to calculate the probabilities in the future states. The probability of outcome ![x_i](doc/teximg/tex_img_35_VWLYY.png) in the state ![X_n](doc/teximg/tex_img_36_NEICT.png) using this algorithm is: \begin{equation} P(X_n = x_i) = \sum_{x_j} P(X_n = x_i | X_{n - 1} = x_j) P(X_{n - 1} = x_j) \end{equation} -This can be easily computed for each state because we know the initial probability $P(X_1 = x_k)$ for every $k$ and the transition probabilities for each state can also be calculated using this algorothm. +This can be easily computed for each state because we know the initial probability ![P(X_1 = x_k)](doc/teximg/tex_img_37_QDJSL.png) for every ![k](doc/teximg/tex_img_38_JJ2GB.png) and the transition probabilities for each state can also be calculated using this algorothm. -Now that we have this solution, we can say that the probability that the $n$th day after today will be sunny in our example is: +Now that we have this solution, we can say that the probability that the ![n](doc/teximg/tex_img_39_UJGWV.png)th day after today will be sunny in our example is: \begin{equation} P(X_{n + 1} = sunny) = \sum_{x_j \in \{sunny, rainy\}} P(X_n = sunny | X_{n - 1} = x_j) P(X_{n - 1} = x_j) @@ -216,18 +216,18 @@ Finally, let's take a look at the state of weather with initial observation of r An interesting question to ask in problems modeled using Markov chains is about the state as we continue simulating for a long time. As we simulate further into the future, the Moarkov chain becomes longer and uncertainty is accumulated. -In the general case, we cannot say what the state will be in the far future. However, for most Markov chains, we will eventually end up in the same distribution no matter what the initial distribution is. In other words, the distribution we end up in is independent of the initial distribution. In such cases, the distribution we end up with is called the **stationary distribution** of the chain. This distribution is denoted by and satisfies the condition below. +In the general case, we cannot say what the state will be in the far future. However, for most Markov chains, we will eventually end up in the same distribution no matter what the initial distribution is. In other words, the distribution we end up in is independent of the initial distribution. In such cases, the distribution we end up with is called the **stationary distribution** of the chain. This distribution is denoted by ![P_\infty](doc/teximg/tex_img_40_9PSGY.png) and satisfies the condition below. -
- -
+\begin{equation} + P_{\infty}(X) = P_{\infty + 1}(X) = \sum_{x} P(X|x) P_{\infty}(x) +\end{equation} In the weather example we mentioned earlier, the stationary distribution denotes the probability that each day will be sunny or rainy in general. ## Most Likely Explanation: Mini-Viterbi Algorithm -Sometimes we would like to find out what happened in the past using the knowledge that we have in present. Consider the trip we discussed earlier as an example. Imagine that you and your friends want to go outside, but some of your friends warn you that everything might be wet if it had rained in the past $n$ days. Since your destination is too far, you do not know about its past, so we need to reason about the most likely sequence of your destination's weather in the past $n$ days. The course of this problem can be shown using the following figure. +Sometimes we would like to find out what happened in the past using the knowledge that we have in present. Consider the trip we discussed earlier as an example. Imagine that you and your friends want to go outside, but some of your friends warn you that everything might be wet if it had rained in the past ![n](doc/teximg/tex_img_41_SD0PW.png) days. Since your destination is too far, you do not know about its past, so we need to reason about the most likely sequence of your destination's weather in the past ![n](doc/teximg/tex_img_42_R5HHO.png) days. The course of this problem can be shown using the following figure.
Weather CPT @@ -235,15 +235,14 @@ Sometimes we would like to find out what happened in the past using the knowledg Again, a slow answer would be to enumerate all possible combinations and calculate the probabilities. However, we can calculate the most likely sequence inductively, remember the incremental probabilities, and update the most likely sequence as we proceed. This approach is called the **Mini-Viterbi Algorithm**. -In the Mini-Viterbi algorithm, we define $m_t[x]$ and $a_t[x]$ as: +In the Mini-Viterbi algorithm, we define ![m_t[x]](doc/teximg/tex_img_43_2ID9V.png) and ![a_t[x]](doc/teximg/tex_img_44_24HEH.png) as: -
- -
-
-
- -
+\begin{equation} + m_t[x] = \max_{x_{1:t - 1}} P(x_{1:t - 1}, x) +\end{equation} +\begin{equation} + a_t[x] = arg\!\max_{x_{1:t - 1}} P(x_{1:t - 1}, x) +\end{equation} The equation above can be rewritten to: \begin{equation} @@ -253,9 +252,9 @@ The equation above can be rewritten to: m_t[x] = \max_{x_{1:t - 1}} P(x_t | x_{t - 1}) m_{t - 1}[x] \end{equation} -Using this method, the most likely sequence in the past $t - 1$ states, also known as the most likely explanation, will be available in $a_t[x]$. +Using this method, the most likely sequence in the past ![t - 1](doc/teximg/tex_img_45_BUT19.png) states, also known as the most likely explanation, will be available in ![a_t[x]](doc/teximg/tex_img_46_RF4UD.png). -In our example, we can use the Mini-Viterbi algorithm starting from $x_1 = The\ weather\ n\ days\ ago$, and the most likely sequence would be available in $a_{n + 1}[x_{today}]$. +In our example, we can use the Mini-Viterbi algorithm starting from ![x_1 = The\ weather\ n\ days\ ago](doc/teximg/tex_img_47_C2ALI.png), and the most likely sequence would be available in ![a_{n + 1}[x_{today}]](doc/teximg/tex_img_48_CW388.png). ## Hidden Markov Model Let's start with the following scenario: @@ -270,10 +269,11 @@ Now, we get the time involved in our problem. Suppose that if today is sunny, to Consider the figure below to get a better understanding of the problem discussed above:
-Bayes Net of Markov Model +Bayes Net of Markov Model
Now, we formulate the problem formally: + \begin{equation} {E}_t: Bob's\;mood\;at\;time\;t\;(Observation) \in \{happy, grumpy\}\\ {X}_t: Weather's\;condition\;at\;time\;t\;(State) \in \{sunny, rainy\} @@ -295,18 +295,18 @@ Take a closer look at the figure below to find out HMM's structure:
Regarding the structure above, an HMM is defined by: -- $P(X_1)$ as initial distribution -- $P(X_t \mid X_{t-1})$ as transitions distribution -- $P(E_t \mid X_t)$ as emissions distribution +- ![P(X_1)](doc/teximg/tex_img_49_VS8EU.png) as initial distribution +- ![P(X_t \mid X_{t-1})](doc/teximg/tex_img_50_33EWV.png) as transitions distribution +- ![P(E_t \mid X_t)](doc/teximg/tex_img_51_45OZK.png) as emissions distribution We will dive into more mathematics in the next part. ### Chain Rule and Joint Distribution of Hidden Markov Model -Let's denote $X_t,X_{t-1},\cdots,X_2,X_1$ by $X_{t:1}$. Similarly, denote $E_t,E_{t-1},\cdots,E_2,E_1$ by $E_{t:1}$. +Let's denote ![X_t,X_{t-1},\cdots,X_2,X_1](doc/teximg/tex_img_52_C1ZZ2.png) by ![X_{t:1}](doc/teximg/tex_img_53_OJDUH.png). Similarly, denote ![E_t,E_{t-1},\cdots,E_2,E_1](doc/teximg/tex_img_54_I236P.png) by ![E_{t:1}](doc/teximg/tex_img_55_RBJQS.png). We are going to calculate joint distribution of HMM. -From the Chain Rule, every joint distribution over $X_{t:1}, E_{t:1}$ can be written as: +From the Chain Rule, every joint distribution over ![X_{t:1}, E_{t:1}](doc/teximg/tex_img_56_DVVKF.png) can be written as: \begin{equation} P(X_{t:1}, E_{t:1})=P(X_1)P(E_1 \mid X_1)P(X_2\mid X_1,E_1)P(E_2\mid X_{2:1},E_1)\cdots P(X_t\mid X_{t-1:1},E_{t-1:1})P(E_t\mid X_{t:1},E_{t-1:1}) From b573dc320c088f7c249e00af4a6cf3ee8867b67f Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Sat, 18 Dec 2021 01:04:12 +0330 Subject: [PATCH 09/33] Edit jpg to jpeg --- notebooks/12_temporal_probability_models/index.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/notebooks/12_temporal_probability_models/index.md b/notebooks/12_temporal_probability_models/index.md index 75c975ab..c9cf6a14 100644 --- a/notebooks/12_temporal_probability_models/index.md +++ b/notebooks/12_temporal_probability_models/index.md @@ -42,7 +42,7 @@ So far, we have learned how to deal with probabilistic reasoning in static world Let' start with an example of a static problem: >Imagine that your car has broken down. You may guess that the problem is related to the engine, batteries, or other car parts. You may take the vehicle to a car repair shop to diagnose the problem and fix it. You know that during the diagnosis process, the broken part remains broken, so the state of the car doesn't change.
-car repair +car repair
@@ -51,7 +51,7 @@ But our problems aren't always static. Consider the below example:
-diabetes +diabetes
@@ -269,7 +269,7 @@ Now, we get the time involved in our problem. Suppose that if today is sunny, to Consider the figure below to get a better understanding of the problem discussed above:
-Bayes Net of Markov Model +Bayes Net of Markov Model
Now, we formulate the problem formally: @@ -367,7 +367,7 @@ Hidden Markov Models (HMM) are applied to interoceptive data acquired by a movin In Robot tracking HMMs, observations are sensors' continuous perceptions and corresponding states are positions in the map.
-machine translation +machine translation
## Conclusion From febab340cff8e9fdda083311a35fb6f5fb5b1655 Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Sat, 18 Dec 2021 01:05:56 +0330 Subject: [PATCH 10/33] Edit image --- notebooks/12_temporal_probability_models/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/12_temporal_probability_models/index.md b/notebooks/12_temporal_probability_models/index.md index c9cf6a14..d190f17a 100644 --- a/notebooks/12_temporal_probability_models/index.md +++ b/notebooks/12_temporal_probability_models/index.md @@ -367,7 +367,7 @@ Hidden Markov Models (HMM) are applied to interoceptive data acquired by a movin In Robot tracking HMMs, observations are sensors' continuous perceptions and corresponding states are positions in the map.
-machine translation +machine translation
## Conclusion From 79dafdf4a82582d4e4cbb7a303cb421ce8a61f3b Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Sat, 18 Dec 2021 01:43:45 +0330 Subject: [PATCH 11/33] Change LaTeX to images --- .../12_temporal_probability_models/index.md | 246 +++++++++--------- .../resources/Nkogk1k.png | Bin 0 -> 15856 bytes .../resources/Pj0cugq.png | Bin 0 -> 13031 bytes .../resources/TwkFmB1.png | Bin 0 -> 9534 bytes .../resources/ruzjAcI.png | Bin 0 -> 75265 bytes .../resources/x2fsViH.png | Bin 0 -> 23302 bytes 6 files changed, 122 insertions(+), 124 deletions(-) create mode 100644 notebooks/12_temporal_probability_models/resources/Nkogk1k.png create mode 100644 notebooks/12_temporal_probability_models/resources/Pj0cugq.png create mode 100644 notebooks/12_temporal_probability_models/resources/TwkFmB1.png create mode 100644 notebooks/12_temporal_probability_models/resources/ruzjAcI.png create mode 100644 notebooks/12_temporal_probability_models/resources/x2fsViH.png diff --git a/notebooks/12_temporal_probability_models/index.md b/notebooks/12_temporal_probability_models/index.md index d190f17a..91642cad 100644 --- a/notebooks/12_temporal_probability_models/index.md +++ b/notebooks/12_temporal_probability_models/index.md @@ -20,6 +20,8 @@ Members: Arshan Dalili, Soroush Jahanzad, Mohammad Hejri [Stationary Distribution](#Stationary-Distribution) +[Most Likely Explanation: Mini-Viterbi Algorithm](#Most-Likely-Explanation:-Mini-Viterbi-Algorithm) + [Hidden Markov Model](#Hidden-Markov-Model) * [Chain Rule and Joint Distribution of Hidden Markov Model](#Chain-Rule-and-Joint-Distribution-of-Hidden-Markov-Model) * [Hidden Markov Models Applications](#Hidden-Markov-Model-Applications) @@ -67,136 +69,129 @@ Retake variables of the diabetes example. We know that these variables can chang >_**Hence we can model the diabetes status of a patient using a Temporal Probability Model.**_ -There are some variables like measured blood sugar and pulse rate that are measurable, and we denote them as ![{E}_t](doc/teximg/tex_img_0_39OCG.png) at time ![t](doc/teximg/tex_img_1_YYV34.png). - -On the other hand, we have some unmeasurable variables like stomach content and blood sugar. (Notice that measured blood sugar and blood sugar are considered different variables since measured blood sugar has some errors.) and at time ![t](doc/teximg/tex_img_2_GQ4EP.png), we use ![{X}_t](doc/teximg/tex_img_3_VET8W.png) to denote them. +There are some variables like measured blood sugar and pulse rate that are measurable, and we denote them as at time . +On the other hand, we have some unmeasurable variables like stomach content and blood sugar. (Notice that measured blood sugar and blood sugar are considered different variables since measured blood sugar has some errors.) and at time , we use to denote them. In a nutshell, we have: - -\begin{equation} -{E}_t = Observable\;Variable\;at\;time\;t\\ -{X}_t = NonObservable\;Variable\;at\;time\;t -\end{equation} - + Observable Variable At Time t + Non-Observable Variable At Time t + ### Markov Models -Let's define a random variable ![X](doc/teximg/tex_img_4_8A5SX.png) and call its value at a given time ![t](doc/teximg/tex_img_5_S26DU.png) the state of ![X](doc/teximg/tex_img_6_FQ2GJ.png) and denote it as ![X_t](doc/teximg/tex_img_7_79OU6.png). If in our model the future state of ![X](doc/teximg/tex_img_8_MYNF4.png) depends only on the current state of ![X](doc/teximg/tex_img_9_X5BQ6.png) and not on states that had occurred before, we call it a **_First-order Markov Model_**, or simply a **_Markov Model_**. Bayesian network of this model looks like the below network and is referred to as a **_Markov Chain_**: +Let's define a random variable and call its value at a given time the state of and denote it as . If in our model the future state of depends only on the current state of and not on states that had occurred before, we call it a **_First-order Markov Model_**, or simply a **_Markov Model_**. Bayesian network of this model looks like the below network and is referred to as a **_Markov Chain_**:
Bayes Net of Markov Model
In a Markov chain, we can write the Morkov assumption as: - -\begin{equation} +
+ +
+ +*Note:* We also define to denote a set of variables from to . +
+ +
-*Note:* We also define ![X_{a:b}](doc/teximg/tex_img_10_AYRJS.png) to denote a set of variables from ![X_a](doc/teximg/tex_img_11_BRTR9.png) to ![X_b](doc/teximg/tex_img_12_80OWW.png). - -\begin{equation} -X_{a:b} = X_a, X_{a+1}, \cdots, X_{b-1}, X_b -\end{equation} - -In a similar way, we can use **_![i](doc/teximg/tex_img_13_0CF2J.png)th-order Markov Models_** to model situations in which the future state of ![X](doc/teximg/tex_img_14_9E376.png) depends only on ![i](doc/teximg/tex_img_15_P34IE.png) recent states and not on states that had occrred before. In this case we have: +In a similar way, we can use **_ith-order Markov Models_** to model situations in which the future state of depends only on recent states and not on states that had occrred before. In this case we have: -\begin{equation} -P(X_n=x_n|X_{1:n-1}=x_{1:n-1}) = P(X_n = x|X_{n - i:n - 1}=x_{n - i : n - 1}) -\end{equation} +
+ +
### Joint Distributions of Markov Models -Take ![X](doc/teximg/tex_img_16_XZRN9.png) as a Markov Model and consider its Bayesian network (depicted below). +Take as a Markov Model and consider its Bayesian network (depicted below).
Bayes Net of Markov Model
-We can quickly notice that for every ![1 \lt i \leq n](doc/teximg/tex_img_17_HALC6.png) , if we are given ![X_{i-1}](doc/teximg/tex_img_18_Q8NV3.png), then **![X_i](doc/teximg/tex_img_19_5IG3C.png) is independent of ![X_1, X_2, \cdots, X_{i-2}](doc/teximg/tex_img_20_D73OR.png)** since there is only one inactive path between each of them and ![X_i](doc/teximg/tex_img_21_FOUL8.png). (Regarding the fact that ![X_{i-1}](doc/teximg/tex_img_22_ABEBF.png) is given) +We can quickly notice that for every , if we are given , then **is independent of** since there is only one inactive path between each of them and . (Regarding the fact that is given) So we have: - -\begin{equation} - X_i \perp \!\!\! \perp X_1, X_2, \cdots, X_{i-2} \vert X_{i-1} -\end{equation} - + +
+ +
+ + We define two parameters here: * **Transition Probabilities**: - - \begin{equation} - P(X_t \vert X_{t-1}) - \end{equation} - - It shows how each state is related to its previous state and how states develop. +
+ +
+ +It shows how each state is related to its previous state and how states develop. The **_Stationary assumption_** in Markov models ensures that transition probabilities don't change over time. * **Initial State Probability**: - - \begin{equation} - P(X_1) - \end{equation} - - It is the probability of the initial state (![X_1](doc/teximg/tex_img_23_13HCZ.png)), and we usually know its value. +
+ +
+ +It is the probability of the initial state (), and we usually know its value. + +We can find the joint distribution of regarding the Bayesian network and the parameters we defined. If we consider to , then we have: +
+ +
- We can find the joint distribution of X regarding the Bayesian network and the parameters we defined. If we consider ![X_1](doc/teximg/tex_img_24_FG02S.png) to ![X_n](doc/teximg/tex_img_25_LCZQB.png), then we have: - - \begin{equation} - P(X_1, X_2, \cdots, X_{n-1}, X_n) = P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n-1}\vert X_{n-2})P(X_{n}\vert X_{n-1})\\ \Rightarrow P(X_1, X_2, \cdots, X_{n-1}, X_n) =P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) - \end{equation} - In the next section, we will prove this joint distribution using Chain Rule and independence. ### Chain Rule -Take ![X](doc/teximg/tex_img_26_16ZCN.png) as a Markov Model and consider its Bayesian network (depicted below). +Take as a Markov Model and consider its Bayesian network (depicted below).
Bayes Net of Markov Model
-We know that from Chain Rule, every joint distribution of ![X_1](doc/teximg/tex_img_27_P67XU.png) to ![X_n](doc/teximg/tex_img_28_UIG2L.png) can be written as: - - \begin{equation} - P(X_1, X_2, \cdots, X_{n-1}, X_n) = P(X_1)P(X_2\vert X_1)P(X_3\vert X_1, X_2)\cdots P(X_{n}\vert X_1,X_2,\cdots X_{n-1}) - \end{equation} - +We know that from Chain Rule, every joint distribution of to can be written as: +
+ +
+ Now, due to the independence we saw earlier, we can assume: - -\begin{equation} - X_3 \perp \!\!\! \perp X_1\vert X_2, X_4\perp \!\!\! \perp X_1,X_2\vert X_3, \cdots, X_{n}\perp \!\!\! \perp X_1,X_2\cdots X_{n-2}\vert X_{n-1} -\end{equation} - +
+ +
+ So, after applying independence to Chain Rule, we have: - - \begin{equation} - P(X_1, X_2, \cdots, X_{n-1}, X_n)= P(X_1)P(X_2\vert X_1)P(X_3\vert X_1, X_2)\cdots P(X_{n}\vert X_1,X_2,\cdots X_{n-1})\\=P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n}\vert X_{n-1})\\=P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) - \end{equation} - +
+ +
+ which is the same result we get from the previous section. ## Mini-Forward Algorithm -Imagine you want to go on a trip with your friends. Your friends think that trips in sunny days are more fun, so they ask you to schedule the trip for a sunny day. You know that you cannot be completely certain whether the ![n](doc/teximg/tex_img_29_Z8H25.png)th day after today is going to be sunny or rainy, so what will you do? +Imagine you want to go on a trip with your friends. Your friends think that trips in sunny days are more fun, so they ask you to schedule the trip for a sunny day. You know that you cannot be completely certain whether the th day after today is going to be sunny or rainy, so what will you do? -We can solve this problem easily if we assume a Markov model. Suppose that we have measured the transition probabilities for this model and know that if some day is sunny, the next day will be sunny with probability ![P(X_{n + 1} = sunny | X_{n} = sunny) = 0.9](doc/teximg/tex_img_30_VIR8P.png) and will be rainy otherwise. Similarly, if some day is rainy, the next day will be rainy with probability ![P(X_{n + 1} = rainy | X_{n} = rainy) = 0.7](doc/teximg/tex_img_31_GOEIM.png) and will be sunny otherwise. We also know the initial probability because we know today's weather. +We can solve this problem easily if we assume a Markov model. Suppose that we have measured the transition probabilities for this model and know that if some day is sunny, the next day will be sunny with probability and will be rainy otherwise. Similarly, if some day is rainy, the next day will be rainy with probability and will be sunny otherwise. We also know the initial probability because we know today's weather. -Now that we have a good model, we need an efficient approach to find the probability of each outcome in the ![n](doc/teximg/tex_img_32_HP3H9.png)th day after today. A naive approach would be to enumerate all possible sequences of length ![n](doc/teximg/tex_img_33_GATCE.png) which end in each outcome and then add up their probabilities. This will do the work, but it will require a lot of time and effort and is too slow. A better approach is to store our incremental *belief* about the weather and to update it as we consider proceed twords the ![n](doc/teximg/tex_img_34_8ZF17.png)th day in our calculations. This approach is called the **Mini-Forward Algorithm**. +Now that we have a good model, we need an efficient approach to find the probability of each outcome in the th day after today. A naive approach would be to enumerate all possible sequences of length which end in each outcome and then add up their probabilities. This will do the work, but it will require a lot of time and effort and is too slow. A better approach is to store our incremental *belief* about the weather and to update it as we consider proceed twords the th day in our calculations. This approach is called the **Mini-Forward Algorithm**. -In the Mini-Forward algorithm, we use the probabilities for the initial state and forward simulation to calculate the probabilities in the future states. The probability of outcome ![x_i](doc/teximg/tex_img_35_VWLYY.png) in the state ![X_n](doc/teximg/tex_img_36_NEICT.png) using this algorithm is: +In the Mini-Forward algorithm, we use the probabilities for the initial state and forward simulation to calculate the probabilities in the future states. The probability of outcome in the state using this algorithm is: +
+ +
-\begin{equation} - P(X_n = x_i) = \sum_{x_j} P(X_n = x_i | X_{n - 1} = x_j) P(X_{n - 1} = x_j) -\end{equation} -This can be easily computed for each state because we know the initial probability ![P(X_1 = x_k)](doc/teximg/tex_img_37_QDJSL.png) for every ![k](doc/teximg/tex_img_38_JJ2GB.png) and the transition probabilities for each state can also be calculated using this algorothm. +This can be easily computed for each state because we know the initial probability for every and the transition probabilities for each state can also be calculated using this algorothm. -Now that we have this solution, we can say that the probability that the ![n](doc/teximg/tex_img_39_UJGWV.png)th day after today will be sunny in our example is: +Now that we have this solution, we can say that the probability that the th day after today will be sunny in our example is: +
+ +
-\begin{equation} - P(X_{n + 1} = sunny) = \sum_{x_j \in \{sunny, rainy\}} P(X_n = sunny | X_{n - 1} = x_j) P(X_{n - 1} = x_j) -\end{equation} +
+ +
-\begin{equation} - \Rightarrow P(X_{n + 1} = sunny) = 0.9 \times P(X_n = sunny | X_{n - 1} = sunny) + 0.3 \times P(X_n = sunny | X_{n - 1} = rainy) -\end{equation} Now, let's take a look at the state of weather with initial observation of sun:
@@ -216,18 +211,18 @@ Finally, let's take a look at the state of weather with initial observation of r An interesting question to ask in problems modeled using Markov chains is about the state as we continue simulating for a long time. As we simulate further into the future, the Moarkov chain becomes longer and uncertainty is accumulated. -In the general case, we cannot say what the state will be in the far future. However, for most Markov chains, we will eventually end up in the same distribution no matter what the initial distribution is. In other words, the distribution we end up in is independent of the initial distribution. In such cases, the distribution we end up with is called the **stationary distribution** of the chain. This distribution is denoted by ![P_\infty](doc/teximg/tex_img_40_9PSGY.png) and satisfies the condition below. +In the general case, we cannot say what the state will be in the far future. However, for most Markov chains, we will eventually end up in the same distribution no matter what the initial distribution is. In other words, the distribution we end up in is independent of the initial distribution. In such cases, the distribution we end up with is called the **stationary distribution** of the chain. This distribution is denoted by and satisfies the condition below. -\begin{equation} - P_{\infty}(X) = P_{\infty + 1}(X) = \sum_{x} P(X|x) P_{\infty}(x) -\end{equation} +
+ +
In the weather example we mentioned earlier, the stationary distribution denotes the probability that each day will be sunny or rainy in general. ## Most Likely Explanation: Mini-Viterbi Algorithm -Sometimes we would like to find out what happened in the past using the knowledge that we have in present. Consider the trip we discussed earlier as an example. Imagine that you and your friends want to go outside, but some of your friends warn you that everything might be wet if it had rained in the past ![n](doc/teximg/tex_img_41_SD0PW.png) days. Since your destination is too far, you do not know about its past, so we need to reason about the most likely sequence of your destination's weather in the past ![n](doc/teximg/tex_img_42_R5HHO.png) days. The course of this problem can be shown using the following figure. +Sometimes we would like to find out what happened in the past using the knowledge that we have in present. Consider the trip we discussed earlier as an example. Imagine that you and your friends want to go outside, but some of your friends warn you that everything might be wet if it had rained in the past days. Since your destination is too far, you do not know about its past, so we need to reason about the most likely sequence of your destination's weather in the past days. The course of this problem can be shown using the following figure.
Weather CPT @@ -235,26 +230,27 @@ Sometimes we would like to find out what happened in the past using the knowledg Again, a slow answer would be to enumerate all possible combinations and calculate the probabilities. However, we can calculate the most likely sequence inductively, remember the incremental probabilities, and update the most likely sequence as we proceed. This approach is called the **Mini-Viterbi Algorithm**. -In the Mini-Viterbi algorithm, we define ![m_t[x]](doc/teximg/tex_img_43_2ID9V.png) and ![a_t[x]](doc/teximg/tex_img_44_24HEH.png) as: +In the Mini-Viterbi algorithm, we define and as: -\begin{equation} - m_t[x] = \max_{x_{1:t - 1}} P(x_{1:t - 1}, x) -\end{equation} -\begin{equation} - a_t[x] = arg\!\max_{x_{1:t - 1}} P(x_{1:t - 1}, x) -\end{equation} +
+ +
+
+
+ +
The equation above can be rewritten to: -\begin{equation} - m_1[x] = P(X_1 = x1) -\end{equation} -\begin{equation} - m_t[x] = \max_{x_{1:t - 1}} P(x_t | x_{t - 1}) m_{t - 1}[x] -\end{equation} +
+ +
+
+ +
-Using this method, the most likely sequence in the past ![t - 1](doc/teximg/tex_img_45_BUT19.png) states, also known as the most likely explanation, will be available in ![a_t[x]](doc/teximg/tex_img_46_RF4UD.png). +Using this method, the most likely sequence in the past states, also known as the most likely explanation, will be available in . -In our example, we can use the Mini-Viterbi algorithm starting from ![x_1 = The\ weather\ n\ days\ ago](doc/teximg/tex_img_47_C2ALI.png), and the most likely sequence would be available in ![a_{n + 1}[x_{today}]](doc/teximg/tex_img_48_CW388.png). +In our example, we can use the Mini-Viterbi algorithm starting from , and the most likely sequence would be available in . ## Hidden Markov Model Let's start with the following scenario: @@ -273,11 +269,9 @@ Consider the figure below to get a better understanding of the problem discussed
Now, we formulate the problem formally: - -\begin{equation} -{E}_t: Bob's\;mood\;at\;time\;t\;(Observation) \in \{happy, grumpy\}\\ -{X}_t: Weather's\;condition\;at\;time\;t\;(State) \in \{sunny, rainy\} -\end{equation} +
+equation +
So, the question is, how can Alice infer the weather’s condition (state) regarding Bob’s mood (observation)? The problem discussed above is an example of the Hidden Markov Model (HMM). It is not possible to model the problem as a Markov Chain in many problems. So, how can we model these problems? @@ -292,36 +286,40 @@ Take a closer look at the figure below to find out HMM's structure:
Bayes Net of Markov Model
-
+
Regarding the structure above, an HMM is defined by: -- ![P(X_1)](doc/teximg/tex_img_49_VS8EU.png) as initial distribution -- ![P(X_t \mid X_{t-1})](doc/teximg/tex_img_50_33EWV.png) as transitions distribution -- ![P(E_t \mid X_t)](doc/teximg/tex_img_51_45OZK.png) as emissions distribution +- as initial distribution +- as transitions distribution +- as emissions distribution We will dive into more mathematics in the next part. ### Chain Rule and Joint Distribution of Hidden Markov Model -Let's denote ![X_t,X_{t-1},\cdots,X_2,X_1](doc/teximg/tex_img_52_C1ZZ2.png) by ![X_{t:1}](doc/teximg/tex_img_53_OJDUH.png). Similarly, denote ![E_t,E_{t-1},\cdots,E_2,E_1](doc/teximg/tex_img_54_I236P.png) by ![E_{t:1}](doc/teximg/tex_img_55_RBJQS.png). +Let's denote by . Similarly, denote by . We are going to calculate joint distribution of HMM. -From the Chain Rule, every joint distribution over ![X_{t:1}, E_{t:1}](doc/teximg/tex_img_56_DVVKF.png) can be written as: - -\begin{equation} -P(X_{t:1}, E_{t:1})=P(X_1)P(E_1 \mid X_1)P(X_2\mid X_1,E_1)P(E_2\mid X_{2:1},E_1)\cdots P(X_t\mid X_{t-1:1},E_{t-1:1})P(E_t\mid X_{t:1},E_{t-1:1}) -\end{equation} +From the Chain Rule, every joint distribution over can be written as: + +
+equation +
+ Regarding the HMM's structure, we can assume that: - -\begin{equation} -\forall t>1:\;\;\;X_t \perp \!\!\! \perp X_{t-2:1},E_{t-1:1} \mid X_{t-1}\;\;\;and\;\;\;E_t \perp \!\!\! \perp X_{t-1:1},E_{t-1:1} \mid X_{t} -\end{equation} + +
+equation +
+ Recall that every node given its parents is independent of its non-descendants. -Using the properties above, we get: -\begin{equation} -P(X_{t:1}, E_{t:1})=P(X_1)P(E_1 \mid X_1)\prod_{i = 2}^{t} P(X_i \mid X_{i-1})P(E_i \mid X_i) -\end{equation} +Using the properties above, we get: +
+equation +
+ + ### Hidden Markov Model Applications Hidden Markov models are known for their applications in various fields, such as: @@ -367,7 +365,7 @@ Hidden Markov Models (HMM) are applied to interoceptive data acquired by a movin In Robot tracking HMMs, observations are sensors' continuous perceptions and corresponding states are positions in the map.
-machine translation +machine translation
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zn{>4fsyNA_*&`pJV-iXwpY8kQbJbZ}t#?tPu^_6kfCbymBVY%P=P-Y(CsvoaU=yT& zOH?bmS={wFDTSWS=A+C~%?r0@mOEBMf`cthI2q0lIhok)o2so0>jJHSo{S#1sxvUm z*C(ZJ%}oTX-3M}-dUtO)E5}4iO2D7$wBH#SqbdgdQDm8&aZl~m6lmBgiwX+~*ecTn z5y?IvYRJZKyt+a&9nDsggNAsEFBCGY(%6aUQtgsp-j`|O>+jp0hu^Y;sHuALr)lo+bNBIjN8`^_x7_Ag>C!k^O z{i)6>Y=93-FHdi4Nowe|H056tt*f-z@^XesY^04ox;ChsAanIqPk1xGn%`FKw8y1v zhSrx@Zk3T3xNxK&A}p6%8bfgXpduI+Vyp#dq_YI`AP z=wm0`*z|Ld_BCItI?>TSj?=jHG@n%N{J{X>u4)xX4OP2zx~-i=nMAn&PaHI!&UHrd z3LTpe_lj6ya3?1?s_y#Qui^~51`q6oDPSUsTpk7Sw;3Htc3w6#`L)?mJzEva9(_V~ zk{nEModcM!Gz-8u{fQAw7Z@|&9HSHR zj!Q{N9jVc?B~~Eujr^|Abl0p)d-hF%s_5L&fKNLuL%+$Oq%|Ke*nx}|7o$W$TUcZo z1bKYIVg3vVKld|G<&4M9nxq^oT(;u8QGO#uJQOE{UJ$mRTU-Ly7 ztGuv*NYd?DF4@>aH^kL}yWCiJ<^YgJ2G5^Wu7WL(@Al(`{QrIZ`xVFkzWm{A*AL&# ze_pdb^4-e482kRaC9&5gW$$WOqmc#E-|9EFQ|R{kCU%ovHXK{`OedS1FQ=Dh5!Hn literal 0 HcmV?d00001 From 7f158f9229f990017127f716c6723742a18cfaec Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Sat, 18 Dec 2021 01:57:45 +0330 Subject: [PATCH 12/33] Fix LaTeX images --- .../12_temporal_probability_models/index.md | 24 +++++++++--------- .../resources/H103XY7.png | Bin 0 -> 13134 bytes .../resources/ZA2Wszd.png | Bin 0 -> 13859 bytes .../resources/gm6qgk0.png | Bin 0 -> 8798 bytes 4 files changed, 12 insertions(+), 12 deletions(-) create mode 100644 notebooks/12_temporal_probability_models/resources/H103XY7.png create mode 100644 notebooks/12_temporal_probability_models/resources/ZA2Wszd.png create mode 100644 notebooks/12_temporal_probability_models/resources/gm6qgk0.png diff --git a/notebooks/12_temporal_probability_models/index.md b/notebooks/12_temporal_probability_models/index.md index 91642cad..a6cb2e18 100644 --- a/notebooks/12_temporal_probability_models/index.md +++ b/notebooks/12_temporal_probability_models/index.md @@ -20,7 +20,7 @@ Members: Arshan Dalili, Soroush Jahanzad, Mohammad Hejri [Stationary Distribution](#Stationary-Distribution) -[Most Likely Explanation: Mini-Viterbi Algorithm](#Most-Likely-Explanation:-Mini-Viterbi-Algorithm) +[Most Likely Explanation: Mini-Viterbi Algorithm](#Most-Likely-Explanation-Mini-Viterbi-Algorithm) [Hidden Markov Model](#Hidden-Markov-Model) * [Chain Rule and Joint Distribution of Hidden Markov Model](#Chain-Rule-and-Joint-Distribution-of-Hidden-Markov-Model) @@ -71,8 +71,11 @@ Retake variables of the diabetes example. We know that these variables can chang There are some variables like measured blood sugar and pulse rate that are measurable, and we denote them as at time . On the other hand, we have some unmeasurable variables like stomach content and blood sugar. (Notice that measured blood sugar and blood sugar are considered different variables since measured blood sugar has some errors.) and at time , we use to denote them. + In a nutshell, we have: + Observable Variable At Time t + Non-Observable Variable At Time t @@ -85,23 +88,20 @@ Let's define a random variable +
*Note:* We also define to denote a set of variables from to . +
- +
In a similar way, we can use **_ith-order Markov Models_** to model situations in which the future state of depends only on recent states and not on states that had occrred before. In this case we have:
- +
### Joint Distributions of Markov Models @@ -270,7 +270,7 @@ Consider the figure below to get a better understanding of the problem discussed Now, we formulate the problem formally:
-equation +equation
So, the question is, how can Alice infer the weather’s condition (state) regarding Bob’s mood (observation)? @@ -303,20 +303,20 @@ We are going to calculate joint distribution of HMM. From the Chain Rule, every joint distribution over can be written as:
-equation +equation
Regarding the HMM's structure, we can assume that:
-equation +equation
Recall that every node given its parents is independent of its non-descendants. Using the properties above, we get:
-equation +equation
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<57400500+arshandalili@users.noreply.github.com> Date: Tue, 28 Dec 2021 17:46:57 +0330 Subject: [PATCH 13/33] Update index.md --- .../12_temporal_probability_models/index.md | 245 +++++++++--------- 1 file changed, 123 insertions(+), 122 deletions(-) diff --git a/notebooks/12_temporal_probability_models/index.md b/notebooks/12_temporal_probability_models/index.md index a6cb2e18..801f1af8 100644 --- a/notebooks/12_temporal_probability_models/index.md +++ b/notebooks/12_temporal_probability_models/index.md @@ -47,11 +47,9 @@ Let' start with an example of a static problem: car repair
- But our problems aren't always static. Consider the below example: >Take diabetes management of a patient as our problem. We know that, for instance, the patient's blood sugar isn't constant and will vary over time, so there is a dynamic in the variables used to describe the patient's state (such as the blood sugar and insulin). -
diabetes
@@ -68,130 +66,136 @@ As the diabetes example above shows, we need to introduce **time** (or space) fo Retake variables of the diabetes example. We know that these variables can change over time, so to manage the diabetes of the patient, we slice time at specific points and keep track of this evidence to evaluate the current state and predict the future diabetic condition of the patient. >_**Hence we can model the diabetes status of a patient using a Temporal Probability Model.**_ +There are some variables like measured blood sugar and pulse rate that are measurable, and we denote them as ${E}_t$ at time $t$. -There are some variables like measured blood sugar and pulse rate that are measurable, and we denote them as at time . -On the other hand, we have some unmeasurable variables like stomach content and blood sugar. (Notice that measured blood sugar and blood sugar are considered different variables since measured blood sugar has some errors.) and at time , we use to denote them. - +On the other hand, we have some unmeasurable variables like stomach content and blood sugar. (Notice that measured blood sugar and blood sugar are considered different variables since measured blood sugar has some errors.) and at time $t$, we use ${X}_t$ to denote them. In a nutshell, we have: - - Observable Variable At Time t - - Non-Observable Variable At Time t - + +\begin{equation} +{E}_t = Observable\;Variable\;at\;time\;t\\ +{X}_t = NonObservable\;Variable\;at\;time\;t +\end{equation} + ### Markov Models -Let's define a random variable and call its value at a given time the state of and denote it as . If in our model the future state of depends only on the current state of and not on states that had occurred before, we call it a **_First-order Markov Model_**, or simply a **_Markov Model_**. Bayesian network of this model looks like the below network and is referred to as a **_Markov Chain_**: +Let's define a random variable $X$ and call its value at a given time $t$ the state of $X$ and denote it as $X_t$. If in our model the future state of $X$ depends only on the current state of $X$ and not on states that had occurred before, we call it a **_First-order Markov Model_**, or simply a **_Markov Model_**. Bayesian network of this model looks like the below network and is referred to as a **_Markov Chain_**:
Bayes Net of Markov Model
In a Markov chain, we can write the Morkov assumption as: -
- -
-*Note:* We also define to denote a set of variables from to . - -
- -
+\begin{equation} +P(X_n=x_n|X_{1:n-1}=x_{1:n-1}) = P(X_n = x_n|X_{n - 1}=x_{n - 1}) +\end{equation} +*Note:* We also define $X_{a:b}$ to denote a set of variables from $X_a$ to $X_b$. + +\begin{equation} +X_{a:b} = X_a, X_{a+1}, \cdots, X_{b-1}, X_b +\end{equation} + -In a similar way, we can use **_ith-order Markov Models_** to model situations in which the future state of depends only on recent states and not on states that had occrred before. In this case we have: +In a similar way, we can use **_$i$th-order Markov Models_** to model situations in which the future state of $X$ depends only on $i$ recent states and not on states that had occrred before. In this case we have: -
- -
+\begin{equation} +P(X_n=x_n|X_{1:n-1}=x_{1:n-1}) = P(X_n = x|X_{n - i:n - 1}=x_{n - i : n - 1}) +\end{equation} ### Joint Distributions of Markov Models -Take as a Markov Model and consider its Bayesian network (depicted below). +Take $X$ as a Markov Model and consider its Bayesian network (depicted below).
Bayes Net of Markov Model
-We can quickly notice that for every , if we are given , then **is independent of** since there is only one inactive path between each of them and . (Regarding the fact that is given) +We can quickly notice that for every $1\lt i\le n$ , if we are given $X_{i-1}$, then **$X_i$ is independent of $X_1, X_2, \cdots, X_{i-2}$** since there is only one inactive path between each of them and $X_i$. (Regarding the fact that $X_{i-1}$ is given) So we have: - -
- -
- - + +\begin{equation} + X_i \perp \!\!\! \perp X_1, X_2, \cdots, X_{i-2} \vert X_{i-1} +\end{equation} + We define two parameters here: * **Transition Probabilities**: -
- -
- -It shows how each state is related to its previous state and how states develop. + + \begin{equation} + P(X_t \vert X_{t-1}) + \end{equation} + + It shows how each state is related to its previous state and how states develop. The **_Stationary assumption_** in Markov models ensures that transition probabilities don't change over time. * **Initial State Probability**: -
- -
- -It is the probability of the initial state (), and we usually know its value. - -We can find the joint distribution of regarding the Bayesian network and the parameters we defined. If we consider to , then we have: -
- -
+ + \begin{equation} + P(X_1) + \end{equation} + + It is the probability of the initial state ($X_1$), and we usually know its value. + We can find the joint distribution of X regarding the Bayesian network and the parameters we defined. If we consider $X_1$ to $X_n$, then we have: + + \begin{equation} + P(X_1, X_2, \cdots, X_{n-1}, X_n) = P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n-1}\vert X_{n-2})P(X_{n}\vert X_{n-1})\\ \Rightarrow P(X_1, X_2, \cdots, X_{n-1}, X_n) =P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) + \end{equation} + In the next section, we will prove this joint distribution using Chain Rule and independence. ### Chain Rule -Take as a Markov Model and consider its Bayesian network (depicted below). +Take $X$ as a Markov Model and consider its Bayesian network (depicted below).
Bayes Net of Markov Model
-We know that from Chain Rule, every joint distribution of to can be written as: -
- -
- +We know that from Chain Rule, every joint distribution of $X_1$ to $X_n$ can be written as: + + \begin{equation} + P(X_1, X_2, \cdots, X_{n-1}, X_n) = P(X_1)P(X_2\vert X_1)P(X_3\vert X_1, X_2)\cdots P(X_{n}\vert X_1,X_2,\cdots X_{n-1}) + \end{equation} + Now, due to the independence we saw earlier, we can assume: -
- -
- + +\begin{equation} + X_3 \perp \!\!\! \perp X_1\vert X_2, X_4\perp \!\!\! \perp X_1,X_2\vert X_3, \cdots, X_{n}\perp \!\!\! \perp X_1,X_2\cdots X_{n-2}\vert X_{n-1} +\end{equation} + So, after applying independence to Chain Rule, we have: -
- -
- + + \begin{equation} + P(X_1, X_2, \cdots, X_{n-1}, X_n)= P(X_1)P(X_2\vert X_1)P(X_3\vert X_1, X_2)\cdots P(X_{n}\vert X_1,X_2,\cdots X_{n-1})\\=P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n}\vert X_{n-1})\\=P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) + \end{equation} + which is the same result we get from the previous section. ## Mini-Forward Algorithm -Imagine you want to go on a trip with your friends. Your friends think that trips in sunny days are more fun, so they ask you to schedule the trip for a sunny day. You know that you cannot be completely certain whether the th day after today is going to be sunny or rainy, so what will you do? +Imagine you want to go on a trip with your friends. Your friends think that trips in sunny days are more fun, so they ask you to schedule the trip for a sunny day. You know that you cannot be completely certain whether the $n$th day after today is going to be sunny or rainy, so what will you do? -We can solve this problem easily if we assume a Markov model. Suppose that we have measured the transition probabilities for this model and know that if some day is sunny, the next day will be sunny with probability and will be rainy otherwise. Similarly, if some day is rainy, the next day will be rainy with probability and will be sunny otherwise. We also know the initial probability because we know today's weather. +We can solve this problem easily if we assume a Markov model. Suppose that we have measured the transition probabilities for this model and know that if some day is sunny, the next day will be sunny with probability $P(X_{n + 1} = sunny | X_{n} = sunny) = 0.9$ and will be rainy otherwise. Similarly, if some day is rainy, the next day will be rainy with probability $P(X_{n + 1} = rainy | X_{n} = rainy) = 0.7$ and will be sunny otherwise. We also know the initial probability because we know today's weather. -Now that we have a good model, we need an efficient approach to find the probability of each outcome in the th day after today. A naive approach would be to enumerate all possible sequences of length which end in each outcome and then add up their probabilities. This will do the work, but it will require a lot of time and effort and is too slow. A better approach is to store our incremental *belief* about the weather and to update it as we consider proceed twords the th day in our calculations. This approach is called the **Mini-Forward Algorithm**. +Now that we have a good model, we need an efficient approach to find the probability of each outcome in the $n$th day after today. A naive approach would be to enumerate all possible sequences of length $n$ which end in each outcome and then add up their probabilities. This will do the work, but it will require a lot of time and effort and is too slow. A better approach is to store our incremental *belief* about the weather and to update it as we consider proceed twords the $n$th day in our calculations. This approach is called the **Mini-Forward Algorithm**. -In the Mini-Forward algorithm, we use the probabilities for the initial state and forward simulation to calculate the probabilities in the future states. The probability of outcome in the state using this algorithm is: -
- -
+In the Mini-Forward algorithm, we use the probabilities for the initial state and forward simulation to calculate the probabilities in the future states. The probability of outcome $x_i$ in the state $X_n$ using this algorithm is: +\begin{equation} + P(X_n = x_i) = \sum_{x_j} P(X_n = x_i | X_{n - 1} = x_j) P(X_{n - 1} = x_j) +\end{equation} -This can be easily computed for each state because we know the initial probability for every and the transition probabilities for each state can also be calculated using this algorothm. +This can be easily computed for each state because we know the initial probability $P(X_1 = x_k)$ for every $k$ and the transition probabilities for each state can also be calculated using this algorothm. -Now that we have this solution, we can say that the probability that the th day after today will be sunny in our example is: -
- -
+Now that we have this solution, we can say that the probability that the $n$th day after today will be sunny in our example is: -
- -
+\begin{equation} + P(X_{n + 1} = sunny) = \sum_{x_j \in \{sunny, rainy\}} P(X_n = sunny | X_{n - 1} = x_j) P(X_{n - 1} = x_j) +\end{equation} +\begin{equation} + \Rightarrow P(X_{n + 1} = sunny) = 0.9 \times P(X_n = sunny | X_{n - 1} = sunny) + 0.3 \times P(X_n = sunny | X_{n - 1} = rainy) +\end{equation} Now, let's take a look at the state of weather with initial observation of sun:
@@ -211,18 +215,18 @@ Finally, let's take a look at the state of weather with initial observation of r An interesting question to ask in problems modeled using Markov chains is about the state as we continue simulating for a long time. As we simulate further into the future, the Moarkov chain becomes longer and uncertainty is accumulated. -In the general case, we cannot say what the state will be in the far future. However, for most Markov chains, we will eventually end up in the same distribution no matter what the initial distribution is. In other words, the distribution we end up in is independent of the initial distribution. In such cases, the distribution we end up with is called the **stationary distribution** of the chain. This distribution is denoted by and satisfies the condition below. +In the general case, we cannot say what the state will be in the far future. However, for most Markov chains, we will eventually end up in the same distribution no matter what the initial distribution is. In other words, the distribution we end up in is independent of the initial distribution. In such cases, the distribution we end up with is called the **stationary distribution** of the chain. This distribution is denoted by $P_\infty$ and satisfies the condition below. -
- -
+\begin{equation} + P_{\infty}(X) = P_{\infty + 1}(X) = \sum_{x} P(X|x) P_{\infty}(x) +\end{equation} In the weather example we mentioned earlier, the stationary distribution denotes the probability that each day will be sunny or rainy in general. ## Most Likely Explanation: Mini-Viterbi Algorithm -Sometimes we would like to find out what happened in the past using the knowledge that we have in present. Consider the trip we discussed earlier as an example. Imagine that you and your friends want to go outside, but some of your friends warn you that everything might be wet if it had rained in the past days. Since your destination is too far, you do not know about its past, so we need to reason about the most likely sequence of your destination's weather in the past days. The course of this problem can be shown using the following figure. +Sometimes we would like to find out what happened in the past using the knowledge that we have in present. Consider the trip we discussed earlier as an example. Imagine that you and your friends want to go outside, but some of your friends warn you that everything might be wet if it had rained in the past $n$ days. Since your destination is too far, you do not know about its past, so we need to reason about the most likely sequence of your destination's weather in the past $n$ days. The course of this problem can be shown using the following figure.
Weather CPT @@ -230,27 +234,26 @@ Sometimes we would like to find out what happened in the past using the knowledg Again, a slow answer would be to enumerate all possible combinations and calculate the probabilities. However, we can calculate the most likely sequence inductively, remember the incremental probabilities, and update the most likely sequence as we proceed. This approach is called the **Mini-Viterbi Algorithm**. -In the Mini-Viterbi algorithm, we define and as: +In the Mini-Viterbi algorithm, we define $m_t[x]$ and $a_t[x]$ as: -
- -
-
-
- -
+\begin{equation} + m_t[x] = \max_{x_{1:t - 1}} P(x_{1:t - 1}, x) +\end{equation} +\begin{equation} + a_t[x] = arg\!\max_{x_{1:t - 1}} P(x_{1:t - 1}, x) +\end{equation} The equation above can be rewritten to: -
- -
-
- -
+\begin{equation} + m_1[x] = P(X_1 = x1) +\end{equation} +\begin{equation} + m_t[x] = \max_{x_{1:t - 1}} P(x_t | x_{t - 1}) m_{t - 1}[x] +\end{equation} -Using this method, the most likely sequence in the past states, also known as the most likely explanation, will be available in . +Using this method, the most likely sequence in the past $t - 1$ states, also known as the most likely explanation, will be available in $a_t[x]$. -In our example, we can use the Mini-Viterbi algorithm starting from , and the most likely sequence would be available in . +In our example, we can use the Mini-Viterbi algorithm starting from $x_1 = The\ weather\ n\ days\ ago$, and the most likely sequence would be available in $a_{n + 1}[x_{today}]$. ## Hidden Markov Model Let's start with the following scenario: @@ -269,9 +272,11 @@ Consider the figure below to get a better understanding of the problem discussed
Now, we formulate the problem formally: -
-equation -
+ +\begin{equation} +{E}_t: Bob's\;mood\;at\;time\;t\;(Observation) \in \{happy, grumpy\}\\ +{X}_t: Weather's\;condition\;at\;time\;t\;(State) \in \{sunny, rainy\} +\end{equation} So, the question is, how can Alice infer the weather’s condition (state) regarding Bob’s mood (observation)? The problem discussed above is an example of the Hidden Markov Model (HMM). It is not possible to model the problem as a Markov Chain in many problems. So, how can we model these problems? @@ -286,40 +291,36 @@ Take a closer look at the figure below to find out HMM's structure:
Bayes Net of Markov Model
-
+
Regarding the structure above, an HMM is defined by: -- as initial distribution -- as transitions distribution -- as emissions distribution +- $P(X_1)$ as initial distribution +- $P(X_t \mid X_{t-1})$ as transitions distribution +- $P(E_t \mid X_t)$ as emissions distribution We will dive into more mathematics in the next part. ### Chain Rule and Joint Distribution of Hidden Markov Model -Let's denote by . Similarly, denote by . +Let's denote $X_t,X_{t-1},\cdots,X_2,X_1$ by $X_{t:1}$. Similarly, denote $E_t,E_{t-1},\cdots,E_2,E_1$ by $E_{t:1}$. We are going to calculate joint distribution of HMM. -From the Chain Rule, every joint distribution over can be written as: - -
-equation -
- +From the Chain Rule, every joint distribution over $X_{t:1}, E_{t:1}$ can be written as: + +\begin{equation} +P(X_{t:1}, E_{t:1})=P(X_1)P(E_1 \mid X_1)P(X_2\mid X_1,E_1)P(E_2\mid X_{2:1},E_1)\cdots P(X_t\mid X_{t-1:1},E_{t-1:1})P(E_t\mid X_{t:1},E_{t-1:1}) +\end{equation} Regarding the HMM's structure, we can assume that: - -
-equation -
- + +\begin{equation} +\forall t>1:\;\;\;X_t \perp \!\!\! \perp X_{t-2:1},E_{t-1:1} \mid X_{t-1}\;\;\;and\;\;\;E_t \perp \!\!\! \perp X_{t-1:1},E_{t-1:1} \mid X_{t} +\end{equation} Recall that every node given its parents is independent of its non-descendants. -Using the properties above, we get: -
-equation -
- - +Using the properties above, we get: +\begin{equation} +P(X_{t:1}, E_{t:1})=P(X_1)P(E_1 \mid X_1)\prod_{i = 2}^{t} P(X_i \mid X_{i-1})P(E_i \mid X_i) +\end{equation} ### Hidden Markov Model Applications Hidden Markov models are known for their applications in various fields, such as: @@ -385,4 +386,4 @@ of observations. Finally, we introduced some of the HMM's well-known application * [Paulo E. Santos (FEI - Sao Paulo) Slides](https://fei.edu.br/~psantos/slidesIA/CAP15-DBN.pdf) * [CS 188: Artificial Intelligence Lecture Slides, Dan Klein, Spring 2006](https://inst.eecs.berkeley.edu/~cs188/sp06/slides/cs188%20lecture%2018%20--%20HMMs%20(2pp).pdf) -_\*Some conceptual pictures used in this note are under courtesy of [VectorStock Media](https://vectorstock.com)_ \ No newline at end of file +_\*Some conceptual pictures used in this note are under courtesy of [VectorStock Media](https://vectorstock.com)_ From 7146df3fa51dc5a3e13017d4dec4ddaf5310616d Mon Sep 17 00:00:00 2001 From: Arshan Dalili <57400500+arshandalili@users.noreply.github.com> Date: Thu, 30 Dec 2021 18:35:49 +0330 Subject: [PATCH 14/33] Update metadata.yml --- .../12_temporal_probability_models/metadata.yml | 13 +++++++++++-- 1 file changed, 11 insertions(+), 2 deletions(-) diff --git a/notebooks/12_temporal_probability_models/metadata.yml b/notebooks/12_temporal_probability_models/metadata.yml index 555e8568..5a277efc 100644 --- a/notebooks/12_temporal_probability_models/metadata.yml +++ b/notebooks/12_temporal_probability_models/metadata.yml @@ -9,14 +9,23 @@ authors: position: top content: # list of notebook authors - - name: Arshan Dalili, Soroush Jahanzad, Mohammad Hejri # name of author + - name: Arshan Dalili # name of author role: Author # change this if you want contact: # list of contact information - link: https://github.com/arshandalili icon: fab fa-github + - link: mailto:arshandalili@gmail.com + icon: fas fa-envelope + - name: Soroush Jahanzad + role: Author + contact: - link: https://github.com/SJahanzad icon: fab fa-github + + - name: Mohammad Hejri + role: Author + contact: - link: https://github.com/MohammadHejri icon: fab fa-github # optionally add other contact information like @@ -26,4 +35,4 @@ authors: comments: # enable comments for your post label: false - kind: comments \ No newline at end of file + kind: comments From 73aef10b0a2223e36e41afa8d9e2499d4f92555d Mon Sep 17 00:00:00 2001 From: Arshan Dalili <57400500+arshandalili@users.noreply.github.com> Date: Thu, 30 Dec 2021 18:38:55 +0330 Subject: [PATCH 15/33] Update metadata.yml --- .../12_temporal_probability_models/metadata.yml | 17 ++++------------- 1 file changed, 4 insertions(+), 13 deletions(-) diff --git a/notebooks/12_temporal_probability_models/metadata.yml b/notebooks/12_temporal_probability_models/metadata.yml index 5a277efc..0f8a5544 100644 --- a/notebooks/12_temporal_probability_models/metadata.yml +++ b/notebooks/12_temporal_probability_models/metadata.yml @@ -1,18 +1,15 @@ title: Temporal Probability Models # shown on browser tab header: - title: Temporal Probability Models (up to filter/monitor) # title of your notebook - description: Temporal Probability Models and Hidden Markov Models and their Applications # short description of your notebook - + title: Temporal Probability Models (up to filter/monitor) + description: Temporal Probability Models and Hidden Markov Models and their Applications authors: label: position: top content: - # list of notebook authors - - name: Arshan Dalili # name of author - role: Author # change this if you want + - name: Arshan Dalili + role: Author contact: - # list of contact information - link: https://github.com/arshandalili icon: fab fa-github - link: mailto:arshandalili@gmail.com @@ -22,17 +19,11 @@ authors: contact: - link: https://github.com/SJahanzad icon: fab fa-github - - name: Mohammad Hejri role: Author contact: - link: https://github.com/MohammadHejri icon: fab fa-github - # optionally add other contact information like - # - link: # contact link - # icon: # awsomefont tag for link (check: https://fontawesome.com/v5.15/icons) - comments: - # enable comments for your post label: false kind: comments From a8e9fd71ac21c0c639ec7de63045da1d2eaecdfd Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Thu, 30 Dec 2021 22:38:27 +0330 Subject: [PATCH 16/33] Change index.yml --- notebooks/index.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/index.yml b/notebooks/index.yml index d4ccffd3..47fbfb32 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -56,4 +56,4 @@ notebooks: - notebook: notebooks/18_reinforcement_learning/ kind: S2021, LN, Notebook - notebook: notebooks/12_temporal_probability_models/ - kind: S2021, LN, Notebook + kind: F2021, LN, markdown From 3ea5c5ec0adbe6e684261836e1308d3d97ba5e72 Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Thu, 30 Dec 2021 22:41:00 +0330 Subject: [PATCH 17/33] Change index.yml --- notebooks/index.yml | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/index.yml b/notebooks/index.yml index 47fbfb32..f527e46a 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -55,5 +55,5 @@ notebooks: #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ kind: S2021, LN, Notebook - - notebook: notebooks/12_temporal_probability_models/ - kind: F2021, LN, markdown + - markdown: notebooks/12_temporal_probability_models/ + kind: F2021, LN, Markdown From f5db8da5b3e7d2418dcad2c470b3804f73386282 Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Thu, 30 Dec 2021 22:43:30 +0330 Subject: [PATCH 18/33] Change index.yml --- notebooks/index.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/index.yml b/notebooks/index.yml index f527e46a..59b79162 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -55,5 +55,5 @@ notebooks: #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ kind: S2021, LN, Notebook - - markdown: notebooks/12_temporal_probability_models/ + - notebook: notebooks/12_temporal_probability_models/index.md kind: F2021, LN, Markdown From 861f8fececb6cd250e662141739fb7c9ec4a762a Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Thu, 30 Dec 2021 22:46:33 +0330 Subject: [PATCH 19/33] Change index.yml --- notebooks/index.yml | 2 ++ 1 file changed, 2 insertions(+) diff --git a/notebooks/index.yml b/notebooks/index.yml index 59b79162..aad38177 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -57,3 +57,5 @@ notebooks: kind: S2021, LN, Notebook - notebook: notebooks/12_temporal_probability_models/index.md kind: F2021, LN, Markdown + metadata: notebooks/12_temporal_probability_models/index.yml + text: Temporal Probability Models - Part 1 From 34b8a41e64ae6de3c091bbb2060c1329045cf144 Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Fri, 31 Dec 2021 14:02:22 +0330 Subject: [PATCH 20/33] Update index.yml --- notebooks/index.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/index.yml b/notebooks/index.yml index aad38177..355fed80 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -55,7 +55,7 @@ notebooks: #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ kind: S2021, LN, Notebook - - notebook: notebooks/12_temporal_probability_models/index.md + - md: notebooks/12_temporal_probability_models/index.md kind: F2021, LN, Markdown metadata: notebooks/12_temporal_probability_models/index.yml text: Temporal Probability Models - Part 1 From 473cede684daf35abda69f3bc83cabfea92da1fa Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Fri, 31 Dec 2021 14:05:54 +0330 Subject: [PATCH 21/33] Update index.yml --- notebooks/index.yml | 1 - 1 file changed, 1 deletion(-) diff --git a/notebooks/index.yml b/notebooks/index.yml index 355fed80..f592d6de 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -57,5 +57,4 @@ notebooks: kind: S2021, LN, Notebook - md: notebooks/12_temporal_probability_models/index.md kind: F2021, LN, Markdown - metadata: notebooks/12_temporal_probability_models/index.yml text: Temporal Probability Models - Part 1 From f16f738f60fcf8d96e78f319dfedba19d8fb72b6 Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Fri, 31 Dec 2021 14:17:29 +0330 Subject: [PATCH 22/33] Update image names --- .../12_temporal_probability_models/index.md | 26 +++++++++--------- .../{sFAP5bQ.jpeg => car_repair.jpeg} | Bin .../resources/{wvnlp2L.jpeg => diabetes.jpeg} | Bin .../{9tlhpJ9.png => hmm_structure.png} | Bin .../{k4JYqIC.png => markov_bayesnet.png} | Bin .../{lnpogDU.png => sunny_initial.png} | Bin .../{N6fC8TW.png => weather_cpt.png} | Bin ...6zJ.jpeg => weather_mood_cpt_diagram.jpeg} | Bin .../{WmoLVeV.png => weather_state.png} | Bin 9 files changed, 13 insertions(+), 13 deletions(-) rename notebooks/12_temporal_probability_models/resources/{sFAP5bQ.jpeg => car_repair.jpeg} (100%) rename notebooks/12_temporal_probability_models/resources/{wvnlp2L.jpeg => diabetes.jpeg} (100%) rename notebooks/12_temporal_probability_models/resources/{9tlhpJ9.png => hmm_structure.png} (100%) rename notebooks/12_temporal_probability_models/resources/{k4JYqIC.png => markov_bayesnet.png} (100%) rename notebooks/12_temporal_probability_models/resources/{lnpogDU.png => sunny_initial.png} (100%) rename notebooks/12_temporal_probability_models/resources/{N6fC8TW.png => weather_cpt.png} (100%) rename notebooks/12_temporal_probability_models/resources/{YOld6zJ.jpeg => weather_mood_cpt_diagram.jpeg} (100%) rename notebooks/12_temporal_probability_models/resources/{WmoLVeV.png => weather_state.png} (100%) diff --git a/notebooks/12_temporal_probability_models/index.md b/notebooks/12_temporal_probability_models/index.md index 801f1af8..13ef2da6 100644 --- a/notebooks/12_temporal_probability_models/index.md +++ b/notebooks/12_temporal_probability_models/index.md @@ -44,14 +44,14 @@ So far, we have learned how to deal with probabilistic reasoning in static world Let' start with an example of a static problem: >Imagine that your car has broken down. You may guess that the problem is related to the engine, batteries, or other car parts. You may take the vehicle to a car repair shop to diagnose the problem and fix it. You know that during the diagnosis process, the broken part remains broken, so the state of the car doesn't change.
-car repair +car repair
But our problems aren't always static. Consider the below example: >Take diabetes management of a patient as our problem. We know that, for instance, the patient's blood sugar isn't constant and will vary over time, so there is a dynamic in the variables used to describe the patient's state (such as the blood sugar and insulin).
-diabetes +diabetes
@@ -81,7 +81,7 @@ In a nutshell, we have: ### Markov Models Let's define a random variable $X$ and call its value at a given time $t$ the state of $X$ and denote it as $X_t$. If in our model the future state of $X$ depends only on the current state of $X$ and not on states that had occurred before, we call it a **_First-order Markov Model_**, or simply a **_Markov Model_**. Bayesian network of this model looks like the below network and is referred to as a **_Markov Chain_**:
-Bayes Net of Markov Model +Bayes Net of Markov Model
In a Markov chain, we can write the Morkov assumption as: @@ -107,7 +107,7 @@ P(X_n=x_n|X_{1:n-1}=x_{1:n-1}) = P(X_n = x|X_{n - i:n - 1}=x_{n - i : n - 1}) Take $X$ as a Markov Model and consider its Bayesian network (depicted below).
-Bayes Net of Markov Model +Bayes Net of Markov Model
We can quickly notice that for every $1\lt i\le n$ , if we are given $X_{i-1}$, then **$X_i$ is independent of $X_1, X_2, \cdots, X_{i-2}$** since there is only one inactive path between each of them and $X_i$. (Regarding the fact that $X_{i-1}$ is given) @@ -148,7 +148,7 @@ In the next section, we will prove this joint distribution using Chain Rule and ### Chain Rule Take $X$ as a Markov Model and consider its Bayesian network (depicted below).
-Bayes Net of Markov Model +Bayes Net of Markov Model
We know that from Chain Rule, every joint distribution of $X_1$ to $X_n$ can be written as: @@ -199,14 +199,14 @@ Now that we have this solution, we can say that the probability that the $n$th d Now, let's take a look at the state of weather with initial observation of sun:
-Initial Sunny Day +Initial Sunny Day

Finally, let's take a look at the state of weather with initial observation of rain:
-Initial Sunny Day +Weather State
@@ -229,7 +229,7 @@ In the weather example we mentioned earlier, the stationary distribution denotes Sometimes we would like to find out what happened in the past using the knowledge that we have in present. Consider the trip we discussed earlier as an example. Imagine that you and your friends want to go outside, but some of your friends warn you that everything might be wet if it had rained in the past $n$ days. Since your destination is too far, you do not know about its past, so we need to reason about the most likely sequence of your destination's weather in the past $n$ days. The course of this problem can be shown using the following figure.
-Weather CPT +Weather CPT
Again, a slow answer would be to enumerate all possible combinations and calculate the probabilities. However, we can calculate the most likely sequence inductively, remember the incremental probabilities, and update the most likely sequence as we proceed. This approach is called the **Mini-Viterbi Algorithm**. @@ -268,7 +268,7 @@ Now, we get the time involved in our problem. Suppose that if today is sunny, to Consider the figure below to get a better understanding of the problem discussed above:
-Bayes Net of Markov Model +CPT Diagram of Weather and Mood
Now, we formulate the problem formally: @@ -289,7 +289,7 @@ Let’s get into HMM’s structure. We start with a simple Markov chain. At each Take a closer look at the figure below to find out HMM's structure:
-Bayes Net of Markov Model +HMM Structure

@@ -344,7 +344,7 @@ The main core of HMM-based speech recognition systems is the Viterbi algorithm w In speech recognition HMMs, the continuous-valued acoustic signals are our observations. States are particular positions in specific words.
-speech recognition +speech recognition
@@ -357,7 +357,7 @@ Better results can be achieved using attention-based neural machine translation In machine translation HMMs, observations are millions of words and corresponding states are translation options.
-machine translation +machine translation
#### Robot tracking HMMs @@ -366,7 +366,7 @@ Hidden Markov Models (HMM) are applied to interoceptive data acquired by a movin In Robot tracking HMMs, observations are sensors' continuous perceptions and corresponding states are positions in the map.
-machine translation +machine translation
## Conclusion diff --git a/notebooks/12_temporal_probability_models/resources/sFAP5bQ.jpeg b/notebooks/12_temporal_probability_models/resources/car_repair.jpeg similarity index 100% rename from notebooks/12_temporal_probability_models/resources/sFAP5bQ.jpeg rename to notebooks/12_temporal_probability_models/resources/car_repair.jpeg diff --git a/notebooks/12_temporal_probability_models/resources/wvnlp2L.jpeg b/notebooks/12_temporal_probability_models/resources/diabetes.jpeg similarity index 100% rename from notebooks/12_temporal_probability_models/resources/wvnlp2L.jpeg rename to notebooks/12_temporal_probability_models/resources/diabetes.jpeg diff --git a/notebooks/12_temporal_probability_models/resources/9tlhpJ9.png b/notebooks/12_temporal_probability_models/resources/hmm_structure.png similarity index 100% rename from notebooks/12_temporal_probability_models/resources/9tlhpJ9.png rename to notebooks/12_temporal_probability_models/resources/hmm_structure.png diff --git a/notebooks/12_temporal_probability_models/resources/k4JYqIC.png b/notebooks/12_temporal_probability_models/resources/markov_bayesnet.png similarity index 100% rename from notebooks/12_temporal_probability_models/resources/k4JYqIC.png rename to notebooks/12_temporal_probability_models/resources/markov_bayesnet.png diff --git a/notebooks/12_temporal_probability_models/resources/lnpogDU.png b/notebooks/12_temporal_probability_models/resources/sunny_initial.png similarity index 100% rename from notebooks/12_temporal_probability_models/resources/lnpogDU.png rename to notebooks/12_temporal_probability_models/resources/sunny_initial.png diff --git a/notebooks/12_temporal_probability_models/resources/N6fC8TW.png b/notebooks/12_temporal_probability_models/resources/weather_cpt.png similarity index 100% rename from notebooks/12_temporal_probability_models/resources/N6fC8TW.png rename to notebooks/12_temporal_probability_models/resources/weather_cpt.png diff --git a/notebooks/12_temporal_probability_models/resources/YOld6zJ.jpeg b/notebooks/12_temporal_probability_models/resources/weather_mood_cpt_diagram.jpeg similarity index 100% rename from notebooks/12_temporal_probability_models/resources/YOld6zJ.jpeg rename to notebooks/12_temporal_probability_models/resources/weather_mood_cpt_diagram.jpeg diff --git a/notebooks/12_temporal_probability_models/resources/WmoLVeV.png b/notebooks/12_temporal_probability_models/resources/weather_state.png similarity index 100% rename from notebooks/12_temporal_probability_models/resources/WmoLVeV.png rename to notebooks/12_temporal_probability_models/resources/weather_state.png From 89eccbac5b9132eb9f91fd91a9897d6823118730 Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Fri, 31 Dec 2021 14:27:16 +0330 Subject: [PATCH 23/33] Update files --- notebooks/12_temporal_probability_models/index.md | 2 +- ...ation_v3-02-3000x1351.jpeg => localization.jpeg} | Bin notebooks/index.yml | 2 +- 3 files changed, 2 insertions(+), 2 deletions(-) rename notebooks/12_temporal_probability_models/resources/{kapernikov_robot_localization_tutorial_illustration_v3-02-3000x1351.jpeg => localization.jpeg} (100%) diff --git a/notebooks/12_temporal_probability_models/index.md b/notebooks/12_temporal_probability_models/index.md index 13ef2da6..181e20fa 100644 --- a/notebooks/12_temporal_probability_models/index.md +++ b/notebooks/12_temporal_probability_models/index.md @@ -366,7 +366,7 @@ Hidden Markov Models (HMM) are applied to interoceptive data acquired by a movin In Robot tracking HMMs, observations are sensors' continuous perceptions and corresponding states are positions in the map.
-machine translation +robot localization
## Conclusion diff --git a/notebooks/12_temporal_probability_models/resources/kapernikov_robot_localization_tutorial_illustration_v3-02-3000x1351.jpeg b/notebooks/12_temporal_probability_models/resources/localization.jpeg similarity index 100% rename from notebooks/12_temporal_probability_models/resources/kapernikov_robot_localization_tutorial_illustration_v3-02-3000x1351.jpeg rename to notebooks/12_temporal_probability_models/resources/localization.jpeg diff --git a/notebooks/index.yml b/notebooks/index.yml index f592d6de..1248a137 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -55,6 +55,6 @@ notebooks: #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ kind: S2021, LN, Notebook - - md: notebooks/12_temporal_probability_models/index.md + - md: notebooks/12_temporal_probability_models/ kind: F2021, LN, Markdown text: Temporal Probability Models - Part 1 From e3e343ec84ad856c73658eba43220ba38c6828d3 Mon Sep 17 00:00:00 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100644 notebooks/12_temporal_probability_models/resources/x2fsViH.png diff --git a/notebooks/12_temporal_probability_models/doc/index.md b/notebooks/12_temporal_probability_models/doc/index.md deleted file mode 100644 index f5deb14e..00000000 --- a/notebooks/12_temporal_probability_models/doc/index.md +++ /dev/null @@ -1,390 +0,0 @@ -# Temporal Probability Models -## Group Info: -Group Number: 12 -Members: Arshan Dalili, Soroush Jahanzad, Mohammad Hejri - - - - - -## Table of Contents -[Introduction](#Introduction) - -[Reasoning over time or space](#Reasoning-over-time-or-space) -* [What is a Temporal Probability Model?](#What-is-a-Temporal-Probability-Model) -* [Markov Models](#Markov-Models) -* [Joint Distributions of Markov Models](#Joint-Distributions-of-Markov-Models) -* [Chain Rule](#Chain-Rule) - -[Mini-Forward Algorithm](#Mini-Forward-Algorithm) - -[Stationary Distribution](#Stationary-Distribution) - -[Hidden Markov Model](#Hidden-Markov-Model) -* [Chain Rule and Joint Distribution of Hidden Markov Model](#Chain-Rule-and-Joint-Distribution-of-Hidden-Markov-Model) -* [Hidden Markov Models Applications](#Hidden-Markov-Model-Applications) - -[Conclusion](#Conclusion) - -[References](#References) - - - - - -## Introduction -In this lecture note, we are going to talk about **_Temporal Probability Models_**. We will discuss what they are and how they can help us model problems that are subject to change over time. - - -## Reasoning over time or space -So far, we have learned how to deal with probabilistic reasoning in static worlds. That means each random variable has a fixed value and doesn't change over time. However, most of our problems don't have this static behavior, and the state varies over time (or space). - -Let' start with an example of a static problem: ->Imagine that your car has broken down. You may guess that the problem is related to the engine, batteries, or other car parts. You may take the vehicle to a car repair shop to diagnose the problem and fix it. You know that during the diagnosis process, the broken part remains broken, so the state of the car doesn't change. -
-car repair -
- - -But our problems aren't always static. Consider the below example: ->Take diabetes management of a patient as our problem. We know that, for instance, the patient's blood sugar isn't constant and will vary over time, so there is a dynamic in the variables used to describe the patient's state (such as the blood sugar and insulin). - - -
-diabetes -
- - -As you can see, the patient's status varies over time, and our approaches in static probabilistic reasoning won't work for these problems. So we should look for other approaches to solve them. - -Temporal Probability Models can make it easier for us to model and solve these types of problems. - - -### What is a Temporal Probability Model? - -As the diabetes example above shows, we need to introduce **time** (or space) for our models. -Retake variables of the diabetes example. We know that these variables can change over time, so to manage the diabetes of the patient, we slice time at specific points and keep track of this evidence to evaluate the current state and predict the future diabetic condition of the patient. - ->_**Hence we can model the diabetes status of a patient using a Temporal Probability Model.**_ - -There are some variables like measured blood sugar and pulse rate that are measurable, and we denote them as ${E}_t$ at time $t$. - -On the other hand, we have some unmeasurable variables like stomach content and blood sugar. (Notice that measured blood sugar and blood sugar are considered different variables since measured blood sugar has some errors.) and at time $t$, we use ${X}_t$ to denote them. -In a nutshell, we have: - -\begin{equation} -{E}_t = Observable\;Variable\;at\;time\;t\\ -{X}_t = NonObservable\;Variable\;at\;time\;t -\end{equation} - - - -### Markov Models -Let's define a random variable $X$ and call its value at a given time $t$ the state of $X$ and denote it as $X_t$. If in our model the future state of $X$ depends only on the current state of $X$ and not on states that had occurred before, we call it a **_First-order Markov Model_**, or simply a **_Markov Model_**. Bayesian network of this model looks like the below network and is referred to as a **_Markov Chain_**: -
-Bayes Net of Markov Model -
- -In a Markov chain, we can write the Morkov assumption as: - -\begin{equation} -P(X_n=x_n|X_{1:n-1}=x_{1:n-1}) = P(X_n = x_n|X_{n - 1}=x_{n - 1}) -\end{equation} - -*Note:* We also define $X_{a:b}$ to denote a set of variables from $X_a$ to $X_b$. - -\begin{equation} -X_{a:b} = X_a, X_{a+1}, \cdots, X_{b-1}, X_b -\end{equation} - - -In a similar way, we can use **_$i$th-order Markov Models_** to model situations in which the future state of $X$ depends only on $i$ recent states and not on states that had occrred before. In this case we have: - -\begin{equation} -P(X_n=x_n|X_{1:n-1}=x_{1:n-1}) = P(X_n = x|X_{n - i:n - 1}=x_{n - i : n - 1}) -\end{equation} - -### Joint Distributions of Markov Models - -Take $X$ as a Markov Model and consider its Bayesian network (depicted below). -
-Bayes Net of Markov Model -
- -We can quickly notice that for every $1 \lt i \leq n$ , if we are given $X_{i-1}$, then **$X_i$ is independent of $X_1, X_2, \cdots, X_{i-2}$** since there is only one inactive path between each of them and $X_i$. (Regarding the fact that $X_{i-1}$ is given) -So we have: - -\begin{equation} - X_i \perp \!\!\! \perp X_1, X_2, \cdots, X_{i-2} \vert X_{i-1} -\end{equation} - -We define two parameters here: -* **Transition Probabilities**: - - \begin{equation} - P(X_t \vert X_{t-1}) - \end{equation} - - It shows how each state is related to its previous state and how states develop. - The **_Stationary assumption_** in Markov models ensures that transition probabilities don't change over time. -* **Initial State Probability**: - - \begin{equation} - P(X_1) - \end{equation} - - It is the probability of the initial state ($X_1$), and we usually know its value. - - We can find the joint distribution of X regarding the Bayesian network and the parameters we defined. If we consider $X_1$ to $X_n$, then we have: - - \begin{equation} - P(X_1, X_2, \cdots, X_{n-1}, X_n) = P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n-1}\vert X_{n-2})P(X_{n}\vert X_{n-1})\\ \Rightarrow P(X_1, X_2, \cdots, X_{n-1}, X_n) =P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) - \end{equation} - -In the next section, we will prove this joint distribution using Chain Rule and independence. - - - - -### Chain Rule -Take $X$ as a Markov Model and consider its Bayesian network (depicted below). -
-Bayes Net of Markov Model -
- -We know that from Chain Rule, every joint distribution of $X_1$ to $X_n$ can be written as: - - \begin{equation} - P(X_1, X_2, \cdots, X_{n-1}, X_n) = P(X_1)P(X_2\vert X_1)P(X_3\vert X_1, X_2)\cdots P(X_{n}\vert X_1,X_2,\cdots X_{n-1}) - \end{equation} - -Now, due to the independence we saw earlier, we can assume: - -\begin{equation} - X_3 \perp \!\!\! \perp X_1\vert X_2, X_4\perp \!\!\! \perp X_1,X_2\vert X_3, \cdots, X_{n}\perp \!\!\! \perp X_1,X_2\cdots X_{n-2}\vert X_{n-1} -\end{equation} - -So, after applying independence to Chain Rule, we have: - - \begin{equation} - P(X_1, X_2, \cdots, X_{n-1}, X_n)= P(X_1)P(X_2\vert X_1)P(X_3\vert X_1, X_2)\cdots P(X_{n}\vert X_1,X_2,\cdots X_{n-1})\\=P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n}\vert X_{n-1})\\=P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) - \end{equation} - -which is the same result we get from the previous section. - -## Mini-Forward Algorithm - -Imagine you want to go on a trip with your friends. Your friends think that trips in sunny days are more fun, so they ask you to schedule the trip for a sunny day. You know that you cannot be completely certain whether the $n$th day after today is going to be sunny or rainy, so what will you do? - -We can solve this problem easily if we assume a Markov model. Suppose that we have measured the transition probabilities for this model and know that if some day is sunny, the next day will be sunny with probability $P(X_{n + 1} = sunny | X_{n} = sunny) = 0.9$ and will be rainy otherwise. Similarly, if some day is rainy, the next day will be rainy with probability $P(X_{n + 1} = rainy | X_{n} = rainy) = 0.7$ and will be sunny otherwise. We also know the initial probability because we know today's weather. - -Now that we have a good model, we need an efficient approach to find the probability of each outcome in the $n$th day after today. A naive approach would be to enumerate all possible sequences of length $n$ which end in each outcome and then add up their probabilities. This will do the work, but it will require a lot of time and effort and is too slow. A better approach is to store our incremental *belief* about the weather and to update it as we consider proceed twords the $n$th day in our calculations. This approach is called the **Mini-Forward Algorithm**. - -In the Mini-Forward algorithm, we use the probabilities for the initial state and forward simulation to calculate the probabilities in the future states. The probability of outcome $x_i$ in the state $X_n$ using this algorithm is: - -\begin{equation} - P(X_n = x_i) = \sum_{x_j} P(X_n = x_i | X_{n - 1} = x_j) P(X_{n - 1} = x_j) -\end{equation} - -This can be easily computed for each state because we know the initial probability $P(X_1 = x_k)$ for every $k$ and the transition probabilities for each state can also be calculated using this algorothm. - -Now that we have this solution, we can say that the probability that the $n$th day after today will be sunny in our example is: - -\begin{equation} - P(X_{n + 1} = sunny) = \sum_{x_j \in \{sunny, rainy\}} P(X_n = sunny | X_{n - 1} = x_j) P(X_{n - 1} = x_j) -\end{equation} - -\begin{equation} - \Rightarrow P(X_{n + 1} = sunny) = 0.9 \times P(X_n = sunny | X_{n - 1} = sunny) + 0.3 \times P(X_n = sunny | X_{n - 1} = rainy) -\end{equation} - -Now, let's take a look at the state of weather with initial observation of sun: -
-Initial Sunny Day -
-
- -Finally, let's take a look at the state of weather with initial observation of rain: - -
-Initial Sunny Day -
- - - -## Stationary Distribution - -An interesting question to ask in problems modeled using Markov chains is about the state as we continue simulating for a long time. As we simulate further into the future, the Moarkov chain becomes longer and uncertainty is accumulated. - -In the general case, we cannot say what the state will be in the far future. However, for most Markov chains, we will eventually end up in the same distribution no matter what the initial distribution is. In other words, the distribution we end up in is independent of the initial distribution. In such cases, the distribution we end up with is called the **stationary distribution** of the chain. This distribution is denoted by $P_\infty$ and satisfies the condition below. - -\begin{equation} - P_{\infty}(X) = P_{\infty + 1}(X) = \sum_{x} P(X|x) P_{\infty}(x) -\end{equation} - -In the weather example we mentioned earlier, the stationary distribution denotes the probability that each day will be sunny or rainy in general. - - -## Most Likely Explanation: Mini-Viterbi Algorithm - -Sometimes we would like to find out what happened in the past using the knowledge that we have in present. Consider the trip we discussed earlier as an example. Imagine that you and your friends want to go outside, but some of your friends warn you that everything might be wet if it had rained in the past $n$ days. Since your destination is too far, you do not know about its past, so we need to reason about the most likely sequence of your destination's weather in the past $n$ days. The course of this problem can be shown using the following figure. - -
-Weather CPT -
- -Again, a slow answer would be to enumerate all possible combinations and calculate the probabilities. However, we can calculate the most likely sequence inductively, remember the incremental probabilities, and update the most likely sequence as we proceed. This approach is called the **Mini-Viterbi Algorithm**. - -In the Mini-Viterbi algorithm, we define $m_t[x]$ and $a_t[x]$ as: - -\begin{equation} - m_t[x] = \max_{x_{1:t - 1}} P(x_{1:t - 1}, x) -\end{equation} -\begin{equation} - a_t[x] = arg\!\max_{x_{1:t - 1}} P(x_{1:t - 1}, x) -\end{equation} - -The equation above can be rewritten to: -\begin{equation} - m_1[x] = P(X_1 = x1) -\end{equation} -\begin{equation} - m_t[x] = \max_{x_{1:t - 1}} P(x_t | x_{t - 1}) m_{t - 1}[x] -\end{equation} - -Using this method, the most likely sequence in the past $t - 1$ states, also known as the most likely explanation, will be available in $a_t[x]$. - -In our example, we can use the Mini-Viterbi algorithm starting from $x_1 = The\ weather\ n\ days\ ago$, and the most likely sequence would be available in $a_{n + 1}[x_{today}]$. - -## Hidden Markov Model -Let's start with the following scenario: - -We have two friends Alice and Bob. They live very far apart. So, they talk on the phone. Bob has a mood that changes based on the weather. Whenever they talk on the phone, Bob tells Alice his mood. - -Suppose that if the weather is sunny, then Bob is happy with a probability of 80% and grumpy with a probability of 20%. Likewise, if the weather is rainy, Bob is grumpy with a probability of 60% and happy with a probability of 40%. - -Whenever Bob tells Alice that he is happy, she may guess that the weather is sunny regarding the probabilities discussed above. Similarly, if Bob tells Alice that he is grumpy, she may suppose rainy weather. Notice that there may be some exceptions that despite the rainy weather, Bob is happy, or despite the sunny weather, he is grumpy. - -Now, we get the time involved in our problem. Suppose that if today is sunny, tomorrow will be sunny with the probability of 80%, and rainy with the probability of 20%. Likewise, if today is rainy, tomorrow will be rainy with the probability of 60%, and sunny with the probability of 40%. - -Consider the figure below to get a better understanding of the problem discussed above: -
-Bayes Net of Markov Model -
- -Now, we formulate the problem formally: - -\begin{equation} -{E}_t: Bob's\;mood\;at\;time\;t\;(Observation) \in \{happy, grumpy\}\\ -{X}_t: Weather's\;condition\;at\;time\;t\;(State) \in \{sunny, rainy\} -\end{equation} -So, the question is, how can Alice infer the weather’s condition (state) regarding Bob’s mood (observation)? - -The problem discussed above is an example of the Hidden Markov Model (HMM). It is not possible to model the problem as a Markov Chain in many problems. So, how can we model these problems? - -Most of the time, it is possible to use HMM! If the problem has a superior state that is a Markov Process, we can make an assumption about this superior state (i.e. our belief), and update it as we observe the problem’s inputs. - -In the problem discussed before, our belief is weather’s condition that can be updated regarding the observation, Bob’s mood. - -Let’s get into HMM’s structure. We start with a simple Markov chain. At each state, we add new nodes for the inputs of the problem. These nodes are called observation nodes and act like agent sensors. Note that each observation node solely relies on its corresponding state. - -Take a closer look at the figure below to find out HMM's structure: -
-Bayes Net of Markov Model -
-
- -Regarding the structure above, an HMM is defined by: -- $P(X_1)$ as initial distribution -- $P(X_t \mid X_{t-1})$ as transitions distribution -- $P(E_t \mid X_t)$ as emissions distribution - -We will dive into more mathematics in the next part. - -### Chain Rule and Joint Distribution of Hidden Markov Model -Let's denote $X_t,X_{t-1},\cdots,X_2,X_1$ by $X_{t:1}$. Similarly, denote $E_t,E_{t-1},\cdots,E_2,E_1$ by $E_{t:1}$. - -We are going to calculate joint distribution of HMM. - -From the Chain Rule, every joint distribution over $X_{t:1}, E_{t:1}$ can be written as: - -\begin{equation} -P(X_{t:1}, E_{t:1})=P(X_1)P(E_1 \mid X_1)P(X_2\mid X_1,E_1)P(E_2\mid X_{2:1},E_1)\cdots P(X_t\mid X_{t-1:1},E_{t-1:1})P(E_t\mid X_{t:1},E_{t-1:1}) -\end{equation} -Regarding the HMM's structure, we can assume that: - -\begin{equation} -\forall t>1:\;\;\;X_t \perp \!\!\! \perp X_{t-2:1},E_{t-1:1} \mid X_{t-1}\;\;\;and\;\;\;E_t \perp \!\!\! \perp X_{t-1:1},E_{t-1:1} \mid X_{t} -\end{equation} -Recall that every node given its parents is independent of its non-descendants. - -Using the properties above, we get: -\begin{equation} -P(X_{t:1}, E_{t:1})=P(X_1)P(E_1 \mid X_1)\prod_{i = 2}^{t} P(X_i \mid X_{i-1})P(E_i \mid X_i) -\end{equation} - -### Hidden Markov Model Applications -Hidden Markov models are known for their applications in various fields, such as: -- Signal processing -- Information theory -- Pattern recognition -- Statistical mechanics -- Physics -- Chemistry -- Economics -- Finance -- Thermodynamics - - - -Below, some of its popular applications are described in more details: - -#### Speech recognition HMMs -The main core of HMM-based speech recognition systems is the Viterbi algorithm which will be discussed in the following section. The Viterbi algorithm uses dynamic programming to find out the best alignment between the input speech and a given speech model. Nowadays, most Large Vocabulary Continuous Speech Recognition (LVCSR) systems are based on HMMs. - -In speech recognition HMMs, the continuous-valued acoustic signals are our observations. States are particular positions in specific words. - -
-speech recognition -
- - - -#### Machine translation HMMs -A simple approach would be the mechanical substitution of words in one language for words in another. However, this won't result in coherent translations most of the time. - -Better results can be achieved using attention-based neural machine translation (NMT) models. These models focus on specific source positions to produce a translation with considerable improvements over pure encoder-decoder models. To achieve this, We use a Hidden Markov Model (HMM) consisting of neural network-based alignment and lexicon models, which are trained jointly using the forward-backward algorithm. - -In machine translation HMMs, observations are millions of words and corresponding states are translation options. - -
-machine translation -
- -#### Robot tracking HMMs -Hidden Markov Models (HMM) are applied to interoceptive data acquired by a moving robot when contouring an environment. To track the robot accurately, the robot needs a sufficient set of sensors. - -In Robot tracking HMMs, observations are sensors' continuous perceptions and corresponding states are positions in the map. - -
-machine translation -
- -## Conclusion -In this note, we first introduced Temporal Probability Model and discussed why we need them to model problems that vary over time. We then introduce Markov Models, explain their properties, and calculate their joint distribution with Chain Rule. - -Then, we talked about the Mini-Forward algorithm and how it can help us predict the state of the system in the future. After that, we talked about stationary distributions help us to know will eventually end up in the same distribution no matter what the initial distribution is. - -Then, we discussed a popular family of temporal models, called Hidden Markov Models for probabilistic sequence classification. In these models, we link a sequence of observations to a sequence of hidden states that explains those observations. Inference, as our goal, was the process of discovering the sequence of hidden states, given the sequence -of observations. Finally, we introduced some of the HMM's well-known applications. - -## References -* [Artificial Intelligence: A Modern Approach, 4th Edition](http://aima.cs.berkeley.edu/) -* [CE417 Lecture Slides, Prof. Rohabn, Sharif University of Technology](http://ce.sharif.edu/courses/00-01/1/ce417-1/resources/root/Slides/PDF/Session%2015_16.pdf) -* [CSC545 Lecture Slides, University of Miami](https://www.cs.miami.edu/home/visser/csc545-files/ProbabilisticReasoningOverTime.pdf) -* [Speech and Language Processing. Daniel Jurafsky & James H. Martin, Stanford University](https://web.stanford.edu/~jurafsky/slp3/A.pdf) -* [University of Tennesee Lecture Slides](https://nanopdf.com/download/chapter-15-probabilistic-reasoning-over-time_pdf) -* [Paulo E. Santos (FEI - Sao Paulo) Slides](https://fei.edu.br/~psantos/slidesIA/CAP15-DBN.pdf) -* [CS 188: Artificial Intelligence Lecture Slides, Dan Klein, Spring 2006](https://inst.eecs.berkeley.edu/~cs188/sp06/slides/cs188%20lecture%2018%20--%20HMMs%20(2pp).pdf) - -_\*Some conceptual pictures used in this note are under courtesy of [VectorStock Media](https://vectorstock.com)_ \ No newline at end of file diff --git a/notebooks/12_temporal_probability_models/doc/teximg/tex_img_0_39OCG.png b/notebooks/12_temporal_probability_models/doc/teximg/tex_img_0_39OCG.png deleted file mode 100644 index 3130ae498c33cc18005d1bf007d978fd1535a7ed..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 245 zcmeAS@N?(olHy`uVBq!ia0vp^f55H;$Qn~>?A+G=b{|7R6?b;O*5+WcV z;Ns#^R#ujlmUidPosN!<88c?Qd-v|lnKMdCN~>0_dhPN$1*n>{z$3C4XxI@DW}K?c zCk+(L^K@|x;gC&ENHF5`*49j5U_8pkwv5Ffn`=T0LmI1bL4kHde9Oo1{b~B+|Cuy>;Ngh6#<0CmZ-PPMRG*^TmO+ lYyk_)hj)ypniy|evoY*<64vRiI|uR@gQu&X%Q~loCIGYqRDA#d diff 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z{i)6>Y=93-FHdi4Nowe|H056tt*f-z@^XesY^04ox;ChsAanIqPk1xGn%`FKw8y1v zhSrx@Zk3T3xNxK&A}p6%8bfgXpduI+Vyp#dq_YI`AP z=wm0`*z|Ld_BCItI?>TSj?=jHG@n%N{J{X>u4)xX4OP2zx~-i=nMAn&PaHI!&UHrd z3LTpe_lj6ya3?1?s_y#Qui^~51`q6oDPSUsTpk7Sw;3Htc3w6#`L)?mJzEva9(_V~ zk{nEModcM!Gz-8u{fQAw7Z@|&9HSHR zj!Q{N9jVc?B~~Eujr^|Abl0p)d-hF%s_5L&fKNLuL%+$Oq%|Ke*nx}|7o$W$TUcZo z1bKYIVg3vVKld|G<&4M9nxq^oT(;u8QGO#uJQOE{UJ$mRTU-Ly7 ztGuv*NYd?DF4@>aH^kL}yWCiJ<^YgJ2G5^Wu7WL(@Al(`{QrIZ`xVFkzWm{A*AL&# ze_pdb^4-e482kRaC9&5gW$$WOqmc#E-|9EFQ|R{kCU%ovHXK{`OedS1FQ=Dh5!Hn From fa21c369ea514735db9d7ff0abf054afa32664a9 Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Fri, 31 Dec 2021 15:10:34 +0330 Subject: [PATCH 25/33] Update yml config --- notebooks/12_temporal_probability_models/metadata.yml | 2 ++ notebooks/index.yml | 1 + 2 files changed, 3 insertions(+) diff --git a/notebooks/12_temporal_probability_models/metadata.yml b/notebooks/12_temporal_probability_models/metadata.yml index 0f8a5544..e5cdf4cb 100644 --- a/notebooks/12_temporal_probability_models/metadata.yml +++ b/notebooks/12_temporal_probability_models/metadata.yml @@ -19,6 +19,8 @@ authors: contact: - link: https://github.com/SJahanzad icon: fab fa-github + - link: mailto:SoroushJahanzad@gmail.com + icon: fas fa-envelope - name: Mohammad Hejri role: Author contact: diff --git a/notebooks/index.yml b/notebooks/index.yml index 1248a137..a2a84cdd 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -57,4 +57,5 @@ notebooks: kind: S2021, LN, Notebook - md: notebooks/12_temporal_probability_models/ kind: F2021, LN, Markdown + metadata: notebooks/12_temporal_probability_models/metadata.yml text: Temporal Probability Models - Part 1 From ccc436fa6e400d375ecb3c70eda7e5c3b037a25a Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Fri, 31 Dec 2021 15:30:28 +0330 Subject: [PATCH 26/33] Change equation notation to $$ --- .../12_temporal_probability_models/index.md | 92 +++++++++---------- 1 file changed, 46 insertions(+), 46 deletions(-) diff --git a/notebooks/12_temporal_probability_models/index.md b/notebooks/12_temporal_probability_models/index.md index 181e20fa..69351fa7 100644 --- a/notebooks/12_temporal_probability_models/index.md +++ b/notebooks/12_temporal_probability_models/index.md @@ -71,10 +71,10 @@ There are some variables like measured blood sugar and pulse rate that are measu On the other hand, we have some unmeasurable variables like stomach content and blood sugar. (Notice that measured blood sugar and blood sugar are considered different variables since measured blood sugar has some errors.) and at time $t$, we use ${X}_t$ to denote them. In a nutshell, we have: -\begin{equation} +$$ {E}_t = Observable\;Variable\;at\;time\;t\\ {X}_t = NonObservable\;Variable\;at\;time\;t -\end{equation} +$$ @@ -86,22 +86,22 @@ Let's define a random variable $X$ and call its value at a given time $t$ the st In a Markov chain, we can write the Morkov assumption as: -\begin{equation} +$$ P(X_n=x_n|X_{1:n-1}=x_{1:n-1}) = P(X_n = x_n|X_{n - 1}=x_{n - 1}) -\end{equation} +$$ *Note:* We also define $X_{a:b}$ to denote a set of variables from $X_a$ to $X_b$. -\begin{equation} +$$ X_{a:b} = X_a, X_{a+1}, \cdots, X_{b-1}, X_b -\end{equation} +$$ In a similar way, we can use **_$i$th-order Markov Models_** to model situations in which the future state of $X$ depends only on $i$ recent states and not on states that had occrred before. In this case we have: -\begin{equation} +$$ P(X_n=x_n|X_{1:n-1}=x_{1:n-1}) = P(X_n = x|X_{n - i:n - 1}=x_{n - i : n - 1}) -\end{equation} +$$ ### Joint Distributions of Markov Models @@ -113,32 +113,32 @@ Take $X$ as a Markov Model and consider its Bayesian network (depicted below). We can quickly notice that for every $1\lt i\le n$ , if we are given $X_{i-1}$, then **$X_i$ is independent of $X_1, X_2, \cdots, X_{i-2}$** since there is only one inactive path between each of them and $X_i$. (Regarding the fact that $X_{i-1}$ is given) So we have: -\begin{equation} +$$ X_i \perp \!\!\! \perp X_1, X_2, \cdots, X_{i-2} \vert X_{i-1} -\end{equation} +$$ We define two parameters here: * **Transition Probabilities**: - \begin{equation} + $$ P(X_t \vert X_{t-1}) - \end{equation} + $$ It shows how each state is related to its previous state and how states develop. The **_Stationary assumption_** in Markov models ensures that transition probabilities don't change over time. * **Initial State Probability**: - \begin{equation} + $$ P(X_1) - \end{equation} + $$ It is the probability of the initial state ($X_1$), and we usually know its value. We can find the joint distribution of X regarding the Bayesian network and the parameters we defined. If we consider $X_1$ to $X_n$, then we have: - \begin{equation} + $$ P(X_1, X_2, \cdots, X_{n-1}, X_n) = P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n-1}\vert X_{n-2})P(X_{n}\vert X_{n-1})\\ \Rightarrow P(X_1, X_2, \cdots, X_{n-1}, X_n) =P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) - \end{equation} + $$ In the next section, we will prove this joint distribution using Chain Rule and independence. @@ -153,21 +153,21 @@ Take $X$ as a Markov Model and consider its Bayesian network (depicted below). We know that from Chain Rule, every joint distribution of $X_1$ to $X_n$ can be written as: - \begin{equation} + $$ P(X_1, X_2, \cdots, X_{n-1}, X_n) = P(X_1)P(X_2\vert X_1)P(X_3\vert X_1, X_2)\cdots P(X_{n}\vert X_1,X_2,\cdots X_{n-1}) - \end{equation} + $$ Now, due to the independence we saw earlier, we can assume: -\begin{equation} +$$ X_3 \perp \!\!\! \perp X_1\vert X_2, X_4\perp \!\!\! \perp X_1,X_2\vert X_3, \cdots, X_{n}\perp \!\!\! \perp X_1,X_2\cdots X_{n-2}\vert X_{n-1} -\end{equation} +$$ So, after applying independence to Chain Rule, we have: - \begin{equation} + $$ P(X_1, X_2, \cdots, X_{n-1}, X_n)= P(X_1)P(X_2\vert X_1)P(X_3\vert X_1, X_2)\cdots P(X_{n}\vert X_1,X_2,\cdots X_{n-1})\\=P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n}\vert X_{n-1})\\=P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) - \end{equation} + $$ which is the same result we get from the previous section. @@ -181,21 +181,21 @@ Now that we have a good model, we need an efficient approach to find the probabi In the Mini-Forward algorithm, we use the probabilities for the initial state and forward simulation to calculate the probabilities in the future states. The probability of outcome $x_i$ in the state $X_n$ using this algorithm is: -\begin{equation} +$$ P(X_n = x_i) = \sum_{x_j} P(X_n = x_i | X_{n - 1} = x_j) P(X_{n - 1} = x_j) -\end{equation} +$$ This can be easily computed for each state because we know the initial probability $P(X_1 = x_k)$ for every $k$ and the transition probabilities for each state can also be calculated using this algorothm. Now that we have this solution, we can say that the probability that the $n$th day after today will be sunny in our example is: -\begin{equation} +$$ P(X_{n + 1} = sunny) = \sum_{x_j \in \{sunny, rainy\}} P(X_n = sunny | X_{n - 1} = x_j) P(X_{n - 1} = x_j) -\end{equation} +$$ -\begin{equation} +$$ \Rightarrow P(X_{n + 1} = sunny) = 0.9 \times P(X_n = sunny | X_{n - 1} = sunny) + 0.3 \times P(X_n = sunny | X_{n - 1} = rainy) -\end{equation} +$$ Now, let's take a look at the state of weather with initial observation of sun:
@@ -217,9 +217,9 @@ An interesting question to ask in problems modeled using Markov chains is about In the general case, we cannot say what the state will be in the far future. However, for most Markov chains, we will eventually end up in the same distribution no matter what the initial distribution is. In other words, the distribution we end up in is independent of the initial distribution. In such cases, the distribution we end up with is called the **stationary distribution** of the chain. This distribution is denoted by $P_\infty$ and satisfies the condition below. -\begin{equation} +$$ P_{\infty}(X) = P_{\infty + 1}(X) = \sum_{x} P(X|x) P_{\infty}(x) -\end{equation} +$$ In the weather example we mentioned earlier, the stationary distribution denotes the probability that each day will be sunny or rainy in general. @@ -236,20 +236,20 @@ Again, a slow answer would be to enumerate all possible combinations and calcula In the Mini-Viterbi algorithm, we define $m_t[x]$ and $a_t[x]$ as: -\begin{equation} +$$ m_t[x] = \max_{x_{1:t - 1}} P(x_{1:t - 1}, x) -\end{equation} -\begin{equation} +$$ +$$ a_t[x] = arg\!\max_{x_{1:t - 1}} P(x_{1:t - 1}, x) -\end{equation} +$$ The equation above can be rewritten to: -\begin{equation} +$$ m_1[x] = P(X_1 = x1) -\end{equation} -\begin{equation} +$$ +$$ m_t[x] = \max_{x_{1:t - 1}} P(x_t | x_{t - 1}) m_{t - 1}[x] -\end{equation} +$$ Using this method, the most likely sequence in the past $t - 1$ states, also known as the most likely explanation, will be available in $a_t[x]$. @@ -273,10 +273,10 @@ Consider the figure below to get a better understanding of the problem discussed Now, we formulate the problem formally: -\begin{equation} +$$ {E}_t: Bob's\;mood\;at\;time\;t\;(Observation) \in \{happy, grumpy\}\\ {X}_t: Weather's\;condition\;at\;time\;t\;(State) \in \{sunny, rainy\} -\end{equation} +$$ So, the question is, how can Alice infer the weather’s condition (state) regarding Bob’s mood (observation)? The problem discussed above is an example of the Hidden Markov Model (HMM). It is not possible to model the problem as a Markov Chain in many problems. So, how can we model these problems? @@ -307,20 +307,20 @@ We are going to calculate joint distribution of HMM. From the Chain Rule, every joint distribution over $X_{t:1}, E_{t:1}$ can be written as: -\begin{equation} +$$ P(X_{t:1}, E_{t:1})=P(X_1)P(E_1 \mid X_1)P(X_2\mid X_1,E_1)P(E_2\mid X_{2:1},E_1)\cdots P(X_t\mid X_{t-1:1},E_{t-1:1})P(E_t\mid X_{t:1},E_{t-1:1}) -\end{equation} +$$ Regarding the HMM's structure, we can assume that: -\begin{equation} +$$ \forall t>1:\;\;\;X_t \perp \!\!\! \perp X_{t-2:1},E_{t-1:1} \mid X_{t-1}\;\;\;and\;\;\;E_t \perp \!\!\! \perp X_{t-1:1},E_{t-1:1} \mid X_{t} -\end{equation} +$$ Recall that every node given its parents is independent of its non-descendants. Using the properties above, we get: -\begin{equation} +$$ P(X_{t:1}, E_{t:1})=P(X_1)P(E_1 \mid X_1)\prod_{i = 2}^{t} P(X_i \mid X_{i-1})P(E_i \mid X_i) -\end{equation} +$$ ### Hidden Markov Model Applications Hidden Markov models are known for their applications in various fields, such as: From 696db082c30467a071f8dcb03840c4f2555ac3e3 Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Fri, 31 Dec 2021 23:18:46 +0330 Subject: [PATCH 27/33] Add a few more examples --- .../12_temporal_probability_models/index.md | 27 ++++++++++++++++-- .../resources/facial_expressions.jpg | Bin 0 -> 43392 bytes 2 files changed, 24 insertions(+), 3 deletions(-) create mode 100644 notebooks/12_temporal_probability_models/resources/facial_expressions.jpg diff --git a/notebooks/12_temporal_probability_models/index.md b/notebooks/12_temporal_probability_models/index.md index 69351fa7..4f1325cd 100644 --- a/notebooks/12_temporal_probability_models/index.md +++ b/notebooks/12_temporal_probability_models/index.md @@ -143,12 +143,10 @@ We define two parameters here: In the next section, we will prove this joint distribution using Chain Rule and independence. - - ### Chain Rule Take $X$ as a Markov Model and consider its Bayesian network (depicted below).
-Bayes Net of Markov Model +Bayes Net of Markov Model
We know that from Chain Rule, every joint distribution of $X_1$ to $X_n$ can be written as: @@ -171,6 +169,18 @@ So, after applying independence to Chain Rule, we have: which is the same result we get from the previous section. +To better grasp the idea, think of a situation in which you want to invest in the stock market. To evaluate your chances of profiting in the short term, you can model the market using a Markov chain where each state $X_t$ can indicate that your investment is "profitable," "indifferent," or "loss-making" in time $t$ compared to $t - 1$. + +While this model can be inaccurate, it can give a preliminary estimate and help you decide better. It can have a quite acceptable performance if we have good estimations of the transition probabilities. Now that we have modeled the situation with a Markov chain, we can use the chain rule to estimate the probability of the investment being profitable in the next $n$ days. This probability can be written as follows. Note that the state of the first day, or $X_0$, is known at the time of investment. + + +$$ +P(X_0=x_0, X_1 = profitable, X_2 = profitable, \cdots, X_n = profitable) +$$ +$$= P(X_0 = x_0) P(X_1 = profitable | X_0 = x_0) \prod_{t=2}^{n}P(X_t = profitable\vert X_{t-1} = profitable) +$$ + + ## Mini-Forward Algorithm Imagine you want to go on a trip with your friends. Your friends think that trips in sunny days are more fun, so they ask you to schedule the trip for a sunny day. You know that you cannot be completely certain whether the $n$th day after today is going to be sunny or rainy, so what will you do? @@ -369,6 +379,17 @@ In Robot tracking HMMs, observations are sensors' continuous perceptions and cor robot localization
+#### Characterizing facial expressions +Facial Action Coding Systems (FACS) prescribes one of the most comprehensive means to characterize facial expressions. + +Now, you may wonder what is the role of HMM in these systems? + +We model this problem as an HMM where observations are the feature vectors extracted from face videos, and corresponding states are the hidden emotional state of the individual. + +
+Facial Expressions +
+ ## Conclusion In this note, we first introduced Temporal Probability Model and discussed why we need them to model problems that vary over time. 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IQT1BI`S^BDnve--jLeEIn*hwV( zU1JnFI(~?bj=%gTI{t+9V<@<%1{N2FgPFAI;fRx!(Ooau^e8Q9?A_G!uhlU*ZCnh4 zb+)%fW64+rp*V1cY(8g`xW)GtULoL~H*5Q?l`S<>p5I0VmYh5Gqwh&^nax!x+X6Xo zFMky7OEzdfqrLP`Ss*C}XdiChY~C51?msGYkRE)nZT}lEs5E?EUZyGY@ZjwU-lglK z_h(1CV@K!|!`W#V(`hU)(zpI*Kpk|+Hfl~omD<-Zd^@psm}^+)DH_(g=-!<6xO8nx zL?GR#S*ghN4e2-9A+WurC1e}t$5Ta4pO>JT7c8ezBuR0K!pNR085)-_nbvG@V<=^1 z`=Y}Y*~?`$_L!l;rxp=?O@22Mk3?yTBjtBj_h#<4nyQQl`Z?wox#Qn|R#j1z;EB{H z*~hSZyL4;~7AXF6pOiQnw!oSb!|&~+zX9I7<95w)mgFY~pWmk;`=G-Z1LJ#B+Y+Bw z&wp>+*6lw6d;fSkAz_JbW{Ws?TE8^S_z6Bkf((tD)As+z=m5Pdxe>^FJa^NFs$tBmr=ZtPqXt(&6b>=4h`gR zv%;*bqJF{#X>Q=}5o~n*Xtl&R@(rfMJ@MU^NvVHQcsc*(hW{J?9m*q1v&Tv|FH+eL zE~D0^DX6q*%I`SX*V4;^i_&02qU;N=fr=UhDTZ;tI_Lu&#F)1++kY8I2Rb$!2-TI# znelK%Jeh&xkacivxW}Y<=NHa^CA^!k4r)X!Y*k9~P(3&>cE>`fC7zS~P=ztjzc-Ow zDen(MF~VV^cz_bstF~Dgt|eY{3Pwf%Gz*icF@zC)31dW|Oz*=O*a2>J8fo&G;Ult& zQav)liK)YK0IjdT0UhZY{S|!*rcPN`pX_)XgG+MKjpHTIlO7`ZKY7dlnl#7n>Hh)j Cl_~N7 literal 0 HcmV?d00001 From a5008a7dd8ed70f7856bc5f9023e27caa3909ff3 Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Fri, 31 Dec 2021 23:26:10 +0330 Subject: [PATCH 28/33] Update index.md --- .../12_temporal_probability_models/index.md | 39 ++++++++++--------- 1 file changed, 21 insertions(+), 18 deletions(-) diff --git a/notebooks/12_temporal_probability_models/index.md b/notebooks/12_temporal_probability_models/index.md index 4f1325cd..7c345495 100644 --- a/notebooks/12_temporal_probability_models/index.md +++ b/notebooks/12_temporal_probability_models/index.md @@ -10,9 +10,9 @@ Members: Arshan Dalili, Soroush Jahanzad, Mohammad Hejri ## Table of Contents [Introduction](#Introduction) -[Reasoning over time or space](#Reasoning-over-time-or-space) -* [What is a Temporal Probability Model?](#What-is-a-Temporal-Probability-Model) -* [Markov Models](#Markov-Models) +[What is a Temporal Probability Model?](#What-is-a-Temporal-Probability-Model) + +[Markov Models](#Markov-Models) * [Joint Distributions of Markov Models](#Joint-Distributions-of-Markov-Models) * [Chain Rule](#Chain-Rule) @@ -37,8 +37,6 @@ Members: Arshan Dalili, Soroush Jahanzad, Mohammad Hejri ## Introduction In this lecture note, we are going to talk about **_Temporal Probability Models_**. We will discuss what they are and how they can help us model problems that are subject to change over time. - -## Reasoning over time or space So far, we have learned how to deal with probabilistic reasoning in static worlds. That means each random variable has a fixed value and doesn't change over time. However, most of our problems don't have this static behavior, and the state varies over time (or space). Let' start with an example of a static problem: @@ -47,9 +45,11 @@ Let' start with an example of a static problem: car repair
+ But our problems aren't always static. Consider the below example: >Take diabetes management of a patient as our problem. We know that, for instance, the patient's blood sugar isn't constant and will vary over time, so there is a dynamic in the variables used to describe the patient's state (such as the blood sugar and insulin). +
diabetes
@@ -60,12 +60,13 @@ As you can see, the patient's status varies over time, and our approaches in sta Temporal Probability Models can make it easier for us to model and solve these types of problems. -### What is a Temporal Probability Model? +## What is a Temporal Probability Model? As the diabetes example above shows, we need to introduce **time** (or space) for our models. Retake variables of the diabetes example. We know that these variables can change over time, so to manage the diabetes of the patient, we slice time at specific points and keep track of this evidence to evaluate the current state and predict the future diabetic condition of the patient. >_**Hence we can model the diabetes status of a patient using a Temporal Probability Model.**_ + There are some variables like measured blood sugar and pulse rate that are measurable, and we denote them as ${E}_t$ at time $t$. On the other hand, we have some unmeasurable variables like stomach content and blood sugar. (Notice that measured blood sugar and blood sugar are considered different variables since measured blood sugar has some errors.) and at time $t$, we use ${X}_t$ to denote them. @@ -78,8 +79,8 @@ $$ -### Markov Models -Let's define a random variable $X$ and call its value at a given time $t$ the state of $X$ and denote it as $X_t$. If in our model the future state of $X$ depends only on the current state of $X$ and not on states that had occurred before, we call it a **_First-order Markov Model_**, or simply a **_Markov Model_**. Bayesian network of this model looks like the below network and is referred to as a **_Markov Chain_**: +## Markov Models +Let's define a random variable $X$ and call its value at a given time $t$ the _**state of $X$**_ and denote it as $X_t$. If in our model, the future state of $X$ depends only on the current state of $X$ and not on states that had occurred before, we call it a **_First-order Markov Model_**, or simply a **_Markov Model_**. Bayesian network of this model looks like the below network and is referred to as a **_Markov Chain_**:
Bayes Net of Markov Model
@@ -110,7 +111,7 @@ Take $X$ as a Markov Model and consider its Bayesian network (depicted below). Bayes Net of Markov Model -We can quickly notice that for every $1\lt i\le n$ , if we are given $X_{i-1}$, then **$X_i$ is independent of $X_1, X_2, \cdots, X_{i-2}$** since there is only one inactive path between each of them and $X_i$. (Regarding the fact that $X_{i-1}$ is given) +We can quickly notice that for every $1 \lt i \leq n$ , if we are given $X_{i-1}$, then **$X_i$ is independent of $X_1, X_2, \cdots, X_{i-2}$** since there is only one inactive path between each of them and $X_i$. (Regarding the fact that $X_{i-1}$ is given) So we have: $$ @@ -134,19 +135,21 @@ We define two parameters here: It is the probability of the initial state ($X_1$), and we usually know its value. - We can find the joint distribution of X regarding the Bayesian network and the parameters we defined. If we consider $X_1$ to $X_n$, then we have: +We can find the joint distribution of X regarding the Bayesian network and the parameters we defined. If we consider $X_1$ to $X_n$, then we have: $$ P(X_1, X_2, \cdots, X_{n-1}, X_n) = P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n-1}\vert X_{n-2})P(X_{n}\vert X_{n-1})\\ \Rightarrow P(X_1, X_2, \cdots, X_{n-1}, X_n) =P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) $$ + In the next section, we will prove this joint distribution using Chain Rule and independence. + ### Chain Rule Take $X$ as a Markov Model and consider its Bayesian network (depicted below).
-Bayes Net of Markov Model +Bayes Net of Markov Model
We know that from Chain Rule, every joint distribution of $X_1$ to $X_n$ can be written as: @@ -173,13 +176,12 @@ To better grasp the idea, think of a situation in which you want to invest in th While this model can be inaccurate, it can give a preliminary estimate and help you decide better. It can have a quite acceptable performance if we have good estimations of the transition probabilities. Now that we have modeled the situation with a Markov chain, we can use the chain rule to estimate the probability of the investment being profitable in the next $n$ days. This probability can be written as follows. Note that the state of the first day, or $X_0$, is known at the time of investment. - $$ P(X_0=x_0, X_1 = profitable, X_2 = profitable, \cdots, X_n = profitable) $$ $$= P(X_0 = x_0) P(X_1 = profitable | X_0 = x_0) \prod_{t=2}^{n}P(X_t = profitable\vert X_{t-1} = profitable) $$ - + ## Mini-Forward Algorithm @@ -255,7 +257,7 @@ $$ The equation above can be rewritten to: $$ - m_1[x] = P(X_1 = x1) + m_1[x] = P(X_1 = x_1) $$ $$ m_t[x] = \max_{x_{1:t - 1}} P(x_t | x_{t - 1}) m_{t - 1}[x] @@ -379,6 +381,7 @@ In Robot tracking HMMs, observations are sensors' continuous perceptions and cor robot localization + #### Characterizing facial expressions Facial Action Coding Systems (FACS) prescribes one of the most comprehensive means to characterize facial expressions. @@ -400,11 +403,11 @@ of observations. Finally, we introduced some of the HMM's well-known application ## References * [Artificial Intelligence: A Modern Approach, 4th Edition](http://aima.cs.berkeley.edu/) -* [CE417 Lecture Slides, Prof. Rohabn, Sharif University of Technology](http://ce.sharif.edu/courses/00-01/1/ce417-1/resources/root/Slides/PDF/Session%2015_16.pdf) -* [CSC545 Lecture Slides, University of Miami](https://www.cs.miami.edu/home/visser/csc545-files/ProbabilisticReasoningOverTime.pdf) +* [CE-417 Lecture Slides, Prof. Rohabn, Sharif University of Technology](http://ce.sharif.edu/courses/00-01/1/ce417-1/resources/root/Slides/PDF/Session%2015_16.pdf) +* [CSC-545 Lecture Slides, University of Miami](https://www.cs.miami.edu/home/visser/csc545-files/ProbabilisticReasoningOverTime.pdf) * [Speech and Language Processing. Daniel Jurafsky & James H. Martin, Stanford University](https://web.stanford.edu/~jurafsky/slp3/A.pdf) * [University of Tennesee Lecture Slides](https://nanopdf.com/download/chapter-15-probabilistic-reasoning-over-time_pdf) * [Paulo E. Santos (FEI - Sao Paulo) Slides](https://fei.edu.br/~psantos/slidesIA/CAP15-DBN.pdf) -* [CS 188: Artificial Intelligence Lecture Slides, Dan Klein, Spring 2006](https://inst.eecs.berkeley.edu/~cs188/sp06/slides/cs188%20lecture%2018%20--%20HMMs%20(2pp).pdf) +* [CS-188: Artificial Intelligence Lecture Slides, Dan Klein, UC Berkeley, Spring 2006](https://inst.eecs.berkeley.edu/~cs188/sp06/slides/cs188%20lecture%2018%20--%20HMMs%20(2pp).pdf) -_\*Some conceptual pictures used in this note are under courtesy of [VectorStock Media](https://vectorstock.com)_ +_\*Some conceptual pictures used in this note are under courtesy of [VectorStock Media](https://vectorstock.com)_ \ No newline at end of file From 4e64be6290f17ed721329ac3ff451cce566d9a9d Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Fri, 31 Dec 2021 23:29:18 +0330 Subject: [PATCH 29/33] Update index.md --- notebooks/12_temporal_probability_models/index.md | 3 --- 1 file changed, 3 deletions(-) diff --git a/notebooks/12_temporal_probability_models/index.md b/notebooks/12_temporal_probability_models/index.md index 7c345495..ba5284e9 100644 --- a/notebooks/12_temporal_probability_models/index.md +++ b/notebooks/12_temporal_probability_models/index.md @@ -1,7 +1,4 @@ # Temporal Probability Models -## Group Info: -Group Number: 12 -Members: Arshan Dalili, Soroush Jahanzad, Mohammad Hejri From 4d6f4a95777cec1fc55b19c0305d25618c6e780b Mon Sep 17 00:00:00 2001 From: Vahid Zehtab <33608325+vahidzee@users.noreply.github.com> Date: Sat, 8 Jan 2022 00:25:07 +0330 Subject: [PATCH 30/33] Fix: md address --- notebooks/index.yml | 1 - 1 file changed, 1 deletion(-) diff --git a/notebooks/index.yml b/notebooks/index.yml index a2a84cdd..1248a137 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -57,5 +57,4 @@ notebooks: kind: S2021, LN, Notebook - md: notebooks/12_temporal_probability_models/ kind: F2021, LN, Markdown - metadata: notebooks/12_temporal_probability_models/metadata.yml text: Temporal Probability Models - Part 1 From d71a634cbcc10587ad4ab9e0293356d842da9898 Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Sat, 8 Jan 2022 01:16:30 +0330 Subject: [PATCH 31/33] Update yml files --- .github/workflows/main.yml | 2 +- notebooks/12_temporal_probability_models/metadata.yml | 1 + 2 files changed, 2 insertions(+), 1 deletion(-) diff --git a/.github/workflows/main.yml b/.github/workflows/main.yml index 7e8ef31a..70726084 100644 --- a/.github/workflows/main.yml +++ b/.github/workflows/main.yml @@ -24,5 +24,5 @@ jobs: github_token: ${{ secrets.GITHUB_TOKEN }} enable_jekyll: true allow_empty_commit: true - publish_dir: . + publish_dir: './webified/' exclude_assets: '.github' diff --git a/notebooks/12_temporal_probability_models/metadata.yml b/notebooks/12_temporal_probability_models/metadata.yml index e5cdf4cb..795cd4b1 100644 --- a/notebooks/12_temporal_probability_models/metadata.yml +++ b/notebooks/12_temporal_probability_models/metadata.yml @@ -4,6 +4,7 @@ header: title: Temporal Probability Models (up to filter/monitor) description: Temporal Probability Models and Hidden Markov Models and their Applications authors: + kind: people label: position: top content: From c36fab738bd9442063e397a7b411fe0039ecf5f4 Mon Sep 17 00:00:00 2001 From: Soroush Jahanzad Date: Sat, 8 Jan 2022 01:27:27 +0330 Subject: [PATCH 32/33] Update index.md --- notebooks/12_temporal_probability_models/index.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/notebooks/12_temporal_probability_models/index.md b/notebooks/12_temporal_probability_models/index.md index ba5284e9..e7b7e6fa 100644 --- a/notebooks/12_temporal_probability_models/index.md +++ b/notebooks/12_temporal_probability_models/index.md @@ -70,7 +70,7 @@ On the other hand, we have some unmeasurable variables like stomach content and In a nutshell, we have: $$ -{E}_t = Observable\;Variable\;at\;time\;t\\ +{E}_t = Observable\;Variable\;at\;time\;t$$$$ {X}_t = NonObservable\;Variable\;at\;time\;t $$ @@ -135,7 +135,7 @@ We define two parameters here: We can find the joint distribution of X regarding the Bayesian network and the parameters we defined. If we consider $X_1$ to $X_n$, then we have: $$ - P(X_1, X_2, \cdots, X_{n-1}, X_n) = P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n-1}\vert X_{n-2})P(X_{n}\vert X_{n-1})\\ \Rightarrow P(X_1, X_2, \cdots, X_{n-1}, X_n) =P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) + P(X_1, X_2, \cdots, X_{n-1}, X_n) = P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n-1}\vert X_{n-2})P(X_{n}\vert X_{n-1}$$$$ \Rightarrow P(X_1, X_2, \cdots, X_{n-1}, X_n) =P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) $$ @@ -164,7 +164,7 @@ $$ So, after applying independence to Chain Rule, we have: $$ - P(X_1, X_2, \cdots, X_{n-1}, X_n)= P(X_1)P(X_2\vert X_1)P(X_3\vert X_1, X_2)\cdots P(X_{n}\vert X_1,X_2,\cdots X_{n-1})\\=P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n}\vert X_{n-1})\\=P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) + P(X_1, X_2, \cdots, X_{n-1}, X_n)= P(X_1)P(X_2\vert X_1)P(X_3\vert X_1, X_2)\cdots P(X_{n}\vert X_1,X_2,\cdots X_{n-1})$$$$=P(X_1)P(X_2\vert X_1)P(X_3\vert X_2)\cdots P(X_{n}\vert X_{n-1})$$$$=P(X_1)\prod_{t=2}^{n}P(X_t \vert X_{t-1}) $$ which is the same result we get from the previous section. @@ -283,7 +283,7 @@ Consider the figure below to get a better understanding of the problem discussed Now, we formulate the problem formally: $$ -{E}_t: Bob's\;mood\;at\;time\;t\;(Observation) \in \{happy, grumpy\}\\ +{E}_t: Bob's\;mood\;at\;time\;t\;(Observation) \in \{happy, grumpy\}$$$$ {X}_t: Weather's\;condition\;at\;time\;t\;(State) \in \{sunny, rainy\} $$ So, the question is, how can Alice infer the weather’s condition (state) regarding Bob’s mood (observation)? From 75405b8a927c4f301daa29b96bc25f466080eeca Mon Sep 17 00:00:00 2001 From: Arshan Dalili <57400500+arshandalili@users.noreply.github.com> Date: Thu, 3 Feb 2022 12:38:05 +0330 Subject: [PATCH 33/33] Update index.md --- notebooks/12_temporal_probability_models/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/12_temporal_probability_models/index.md b/notebooks/12_temporal_probability_models/index.md index e7b7e6fa..89f95503 100644 --- a/notebooks/12_temporal_probability_models/index.md +++ b/notebooks/12_temporal_probability_models/index.md @@ -1,4 +1,4 @@ -# Temporal Probability Models +# Temporal Probability Models - Part 1 @@ -407,4 +407,4 @@ of observations. Finally, we introduced some of the HMM's well-known application * [Paulo E. Santos (FEI - Sao Paulo) Slides](https://fei.edu.br/~psantos/slidesIA/CAP15-DBN.pdf) * [CS-188: Artificial Intelligence Lecture Slides, Dan Klein, UC Berkeley, Spring 2006](https://inst.eecs.berkeley.edu/~cs188/sp06/slides/cs188%20lecture%2018%20--%20HMMs%20(2pp).pdf) -_\*Some conceptual pictures used in this note are under courtesy of [VectorStock Media](https://vectorstock.com)_ \ No newline at end of file +_\*Some conceptual pictures used in this note are under courtesy of [VectorStock Media](https://vectorstock.com)_

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