lectures/03_logreg.ipynb

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    "# Logistic Regression\n",
    "\n",
    "We started with basic discrete distributions for single random variables, and then we modeled pairs of categorical variables with continency tables. Here, we build models for predicting categorical responses given several explanatory variables.\n"
   ]
  },
  {
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   "execution_count": 15,
   "id": "9c1cddcd",
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   "source": [
    "# Setup\n",
    "import torch\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "from torch.distributions import Normal, Bernoulli"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "89c16bc1",
   "metadata": {},
   "source": [
    "\n",
    "## Contingency Tables with Binary Responses\n",
    "\n",
    "A special case of contingency table analyses is when $X \\in \\{1, \\ldots, I\\}$ corresponds to a categorical feature (e.g., to which of $I$ groups a data point belongs) and $Y \\in \\{0,1\\}$ is a binary response. \n",
    "\n",
    "<!-- Contingency tables are often used to compare two groups $X \\in \\{0,1\\}$ based on a binary response variables $Y \\in \\{0,1\\}$. The resulting tables are 2x2. \n",
    "The association between $X$ and $Y$ can be summarized with a variety of statistics: the difference of proportions, the relative risk, and the odds ratio. We will focus on the latter.\n",
    "\n",
    "For a Bernoulli random variable with probability $p$, the odds are defined as \n",
    "\\begin{align*}\n",
    "\\Omega = \\frac{p}{1 - p}. \n",
    "\\end{align*}\n",
    "Inversely, $p = \\frac{\\Omega}{\\Omega + 1}$. -->\n",
    "\n",
    "The corresponding tables are $I \\times 2$. Normalizing each row yields a Bernoulli conditional distribution,\n",
    "\\begin{align*}\n",
    "Y \\mid X=i &\\sim \\mathrm{Bern}(\\pi_{1|i}).\n",
    "\\end{align*}\n",
    "where, $\\pi_{1|i} = \\Pr(Y=1 \\mid X=i)$ or, equivalently, $\\pi_{1|i} = \\E[Y \\mid X=i]$.\n",
    "\n",
    "The conditional distributions are the primary objects of interest when our goal is to make predictions. They can also be used to test for independence by testing the homogeneity of conditionals.\n",
    "<!-- \n",
    "To test whether $Y$ is independent of $X$, we can assess whether these conditionals are homogeneous, i.e., the same for all values of $X$. \n",
    "-->\n",
    "\n",
    "<!-- \\pi_{1|i} = \\frac{\\pi_{i1}}{\\pi_{i0} + \\pi_{i1}} = \\Pr(Y=1 \\mid X=i) \\equiv \\E[Y \\mid X=i]. -->\n",
    "\n",
    "<!-- More generally, we can write the conditional distribution for both values of $X$ as,\n",
    "\\begin{align*}\n",
    "Y \\mid X=x &\\sim \\mathrm{Bern}(\\pi(x))\n",
    "\\end{align*}\n",
    "where\n",
    "\\begin{align*}\n",
    "\\pi(x) \n",
    "&= \\pi_{1|1} x + \\pi_{1|0} (1-x) \\\\\n",
    "&= (\\pi_{1|1} - \\pi_{1|0}) x + \\pi_{1|0}\n",
    "\\end{align*}\n",
    "\n",
    "\n",
    "The odds for row $i$ are,\n",
    "\\begin{align*}\n",
    "\\Omega_i = \\frac{\\pi_{1|i}}{1 - \\pi_{1|i}} \n",
    "= \\frac{\\pi_{i1}}{\\pi_{i0}}.\n",
    "\\end{align*}\n",
    "The odds are non-negative, $\\Omega_i \\in [0, \\infty)$. \n",
    "\n",
    "In practice, it is often more convenient to work with the log odds, $\\log \\Pr(Y=1 \\mid X=i) = \\log \\Omega_i$, which are real valued.  -->\n"
   ]
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   "id": "55c509b3",
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   "source": [
    "\n",
    "## One, Two, Many...\n",
    "\n",
    "What if we have more than one feature, $X_1, \\ldots, X_p$? We could construct a multi-dimensional contingency table to estimate the joint distribution over $(X_1,\\ldots, X_p,Y)$, but we quickly encounter the curse of dimensionality: if each feature takes on $K$ values, there are $\\cO(K^D)$ cells in the table.\n",
    "\n",
    "Moreover, what if the features take on continuous values? Then the notion of a finite table no longer makes sense.\n",
    "\n",
    "Instead, let's try to model the conditional distribution directly. Let $Y \\in \\{0,1\\}$ denote a binary response and let $\\mbX \\in \\reals^p$ denote associated covariates. For example, $Y$ could denote whether or not your favorite football team wins their match, and $X$ could represent features of the match like whether its a home or away game, who their opponent is, etc. We will model the conditional distribution of $Y$ given the covariates,\n",
    "\\begin{align*}\n",
    "Y \\mid \\mbX = \\mbx &\\sim \\mathrm{Bern}(\\pi(\\mbx))\n",
    "\\end{align*}\n",
    "where\n",
    "\\begin{align*}\n",
    "\\pi(\\mbx) &= \\Pr(Y = 1 \\mid \\mbX=\\mbx) = \\E[Y \\mid \\mbX=\\mbx].\n",
    "\\end{align*}\n",
    "\n",
    "This is a standard regression setup. The modeling problem boils down to choosing the functional form of $\\pi(\\mbx)$. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8de11d5d",
   "metadata": {},
   "source": [
    "## Linear Regression\n",
    "If you took STATS 305A, you know pretty much everything there is to know about linear regression with continuous response variables, $Y \\in \\reals$. Why don't we just apply that same model to binary responses? Specifically, let,\n",
    "\\begin{align*}\n",
    "\\pi(\\mbx) &= \\mbbeta^\\top \\mbx = \\sum_{j=1}^p \\beta_j x_j. \n",
    "\\end{align*}\n",
    "\n",
    "The obvious problem is that this linear model produces probabilities or expectations $\\pi(\\mbx) \\in \\reals$ instead of just over the valid range $[0,1]$, which is clearly misspecified!\n",
    "\n",
    "<!-- \n",
    "Then we'll just use ordinary least squares (OLS) to estimate $\\hat{\\mbbeta}$. What could go wrong? After all, $\\{0,1\\} \\subset \\reals$... \n",
    "\n",
    "There are a few issues:\n",
    "1. The linear model produces probabilities $\\pi(\\mbx) \\in \\reals$ instead of just over the valid range $[0,1]$, so the model is necessarily misspecified.\n",
    "2. Moreover, the variance of a Bernoulli random variable changes as a function of the probability,\n",
    "    \\begin{align*}\n",
    "    \\Var[Y \\mid \\mbX=\\mbx] = \\pi(\\mbx) (1 - \\pi(\\mbx)),\n",
    "    \\end{align*}\n",
    "    which violates the homoskedasticity assumption under which OLS is optimal. \n",
    "\n",
    "Nevertheless, it's not a totally crazy thing to do. When the estimated probabilities are in an intermediate range (say, 0.3-0.7), the outputs aren't that different from what we obtain with the alternative models below. But we can do better. -->\n"
   ]
  },
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   "cell_type": "markdown",
   "id": "2f827db0",
   "metadata": {},
   "source": [
    "\n",
    "## Logistic Regression\n",
    "\n",
    "The idea is simple: keep the linear part of linear regression, but apply a **mean (aka inverse link) function**, $f: \\reals \\mapsto [0,1]$, to ensure $\\pi(\\mbx)$ returns valid probabilities,\n",
    "\\begin{align*}\n",
    "\\pi(\\mbx) &= f(\\mbbeta^\\top \\mbx).\n",
    "\\end{align*}\n",
    "There are infinitely many squashing nonlinearities that we could choose for $f$, but a particularly attractive choice is the **logistic (aka sigmoid) function**,\n",
    "\\begin{align*}\n",
    "f(a) = \\frac{e^a}{1 + e^a} = \\frac{1}{1 + e^{-a}} \\triangleq \\sigma(a).\n",
    "\\end{align*}"
   ]
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "a = torch.linspace(-4, 4, 100)\n",
    "plt.plot(a, torch.sigmoid(a))\n",
    "plt.grid(True)\n",
    "plt.xlabel(r\"$a$\")\n",
    "plt.ylabel(r\"$\\sigma(a)$\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "522a6afa",
   "metadata": {},
   "source": [
    "One nice feature of the logistic function is that it is monotonically increasing, so increasing in $\\mbbeta^\\top \\mbx$ yield larger probability estimates. That also means that we can invert the sigmoid function. In doing so, we find that the linear component of the model $\\mbbeta^\\top \\mbx$ correspond to the **log odds** of the binary response since the inverse of the sigmoid function is the **logit function**,\n",
    "\\begin{align*}\n",
    "\\mbbeta^\\top \\mbx = \\sigma^{-1}(\\pi(\\mbx)) &= \\log \\frac{\\pi(\\mbx)}{1 - \\pi(\\mbx)}.\n",
    "\\end{align*}\n",
    "Finally, we'll see that the logistic function leads to some simpler mathematical calculations when it comes to parameter estimation and inference.\n",
    "\n",
    "Another common mean function is the Gaussian CDF, and we'll consider that in a later chapter. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d250d025",
   "metadata": {},
   "source": [
    "<!-- \n",
    "## Relationship with Two-Way Contingency Tables\n",
    "\n",
    "Suppose we have a single, binary covariate $X \\in \\{0,1\\}$. In the last chapter, we constructed 2x2 contingency tables for such settings, and we used the log odds ratio to measure the association between $X$ and $Y$. We could do the same thing with logistic regression.\n",
    "\n",
    "First, we need to extend the model with an intercept term,\n",
    "\\begin{align*}\n",
    "\\pi(x) &= \\sigma \\left(\\beta_0 + \\beta_1 x \\right),\n",
    "\\end{align*}\n",
    "for $x \\in \\{0,1\\}$.\n",
    "\n",
    ":::{admonition} Note about intercepts\n",
    ":class: warning\n",
    "We explicitly separated the intercept term above, but in general we can assume that the covariates include a constant term, $\\mbx = (1, x_1, \\ldots, x_p)^\\top \\in \\reals^{p+1}$. Then the first coefficient in $\\mbbeta = (\\beta_0, \\beta_1, \\ldots, \\beta_p)^\\top \\in \\reals^{p+1}$ corresponds to the intercept.\n",
    ":::\n",
    "\n",
    "Under this model, $\\beta_1$ specifies the log odds,\n",
    "\\begin{align*}\n",
    "\\beta_1 = \\sigma^{-1}(\\pi(1)) - \\sigma^{-1}(\\pi(0)) \n",
    "= \\log \\frac{\\pi(1) / (1 - \\pi(1))}{\\pi(0) / (1 - \\pi(0))} \n",
    "= \\log \\theta.\n",
    "\\end{align*}\n",
    "In other words, the coefficients of the logistic regression model correspond to the log odds in a contingency table. -->\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd52571e",
   "metadata": {},
   "source": [
    "## Hacky Parameter Estimation with OLS\n",
    "\n",
    "How do we estimate the parameters, $\\hat{\\mbbeta}$? If the model were linear, then our first inclination might be to use ordinary least squares (OLS). Of course, the sigmoidal function above renders the model nonlinear, but what if we just used a Taylor approximation around the origin,\n",
    "\\begin{align*}\n",
    "\\sigma(a) &\\approx \\sigma(0) + \\sigma'(0) a \n",
    "= \\frac{1}{2} + \\frac{a}{4}\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "ffd4db47",
   "metadata": {
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "image/png": 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x9fXll19+YfLkyaSlpdGnTx/atWvHb7/9xrvvvsuUKVN4/fXXAbjnnnvo06cPd955J6WlpcybN48PP/yQzz77jMDAwHJ89UVEKpHdDqvehY96QtZeCK0NgxdCp2QwmYyurlJVneMaFxtLAbwUW65VzUCYKz/72XTwrXZBU7Rs2ZKWLVs6x88//zyzZ8/m22+/JTk5mVq1apGUlMTUqVNp164dAFOnTuWqq66iXr16J82Xn5/P5MmTmTZtGr179wbggw8+IDU1lY8++oinnnrKue64cePo0aPHGet77rnnnL9PSEjgySefZMaMGQwfPty5vKioiE8++YS4uDgA3n77bfr27cu///1vYmJiAPD19WXKlCkEBgbStGlTxo0bx1NPPcXzzz+P+a9j9w0bNuTVV191zvuvf/2L+Ph43nnnHUwmE40bNyYtLY1nnnmGF154AbPZzHvvvUeLFi145JFHmDVrFmPGjKFNmzbl++KLiFSWwuPwTTJs+84xbnwNXPcOBIQbW5dBtKdJzkteXh5PPvkkl156KWFhYQQFBbF161bnniaAIUOG8Pnnn1NUVERJSQnTp09n8ODBp5xv9+7dWCwWLr/8f/f18PHxoX379mzdurXMum3btj1rfTNnzuTyyy8nJiaGoKAgnnvuuTK1AdSuXdsZmAA6duyIzWYrc95Wy5Yty+z96dixI3l5eezfv9+57J9hZ+vWrXTs2BHT3/4H1qlTJ/Ly8jhw4AAA4eHhfPTRR7z77rvUr1+fZ5555qw9iYhUqv1rYfKVjsDk5Qu9X4VbP71oAxNoT5NxfAIde3zKwWazkZObS0hwsHPvxgV/9gV68sknSU1N5fXXX6dBgwYEBARw0003lTnfp1+/fvj5+TF79mx8fX2xWCzcdNNNF/zZ1aqdeS/ZypUrufPOOxk7dixJSUmEhoYyY8YM/v3vf1/wZ59PPaezbNkyvLy8OHjwIPn5+QQHB7u4MhGR82CzwapJ8MMYsJVCeALcPA1iWxtcmPEUmoxiMpX/EJnNBj5Wx/pV5HLOX375hYEDB3L99dcDjj1Pe/bsKbOOt7c3AwYMYOrUqfj6+nLbbbcREBBwyvnq1q3rPC+oTp06AFgsFtauXctjjz12TrWtWLGCOnXq8K9//cu5bO/evSett2/fPtLT04mNdRwmXbVqFWazmUaNGjnX+e233ygsLHTWvWrVKoKCgoiPjz/t51966aV8/fXX2O12596mFStWEBwcTK1atZzjV155hblz5/L000+TnJzMxx9/fE59ioi4XMExmD0Udix0jJteD/3eBP9QY+uqIhSa5Lw0bNiQWbNm0a9fP0wmEyNHjsRms5203n333cell14KOILW6VSrVo2hQ4fy1FNPERERQe3atXn11VcpKCjg3nvvPefa9u3bx4wZM2jXrh3z5s1j9uzZJ63n7+/PgAEDeP3118nJyeGRRx7hlltucZ7PBI5bCNx7770899xz7Nmzh9GjR5OcnHzGPX4PPfQQEydO5OGHHyY5OZnt27czZswYHnroIcxmM7m5udx999088sgj9O7dm1q1atGuXTv69evn3BM3YsQI0tLS+OSTT86pdxGR87Z3JXx9L+SkgZcf9BoPbQdfdCd7n4lCk5yXCRMmMHjwYDp16kRkZCRPP/00OTk5J63XsGFDOnXqxLFjx0hMTDzjnOPHj8dut3P33XeTm5tL27ZtWbhwIeHh53b8/Nprr+Xxxx8nOTmZ4uJi+vbty8iRI0+6GWeDBg244YYb6NOnD8eOHeOaa67h//7v/8qs061bNxo2bMiVV15JcXExt99++1lv6hkXF8f8+fN56qmnaNmyJREREQwePJgnn3wSgEcffZRq1arx0ksvAdC8eXNeeuklHnjgATp27EhcXBwHDx486RwsEZEKYbPBLxPhxxfAboWI+o7DcTVPfXXyxcxkt7vg+nMBICcnh9DQULKzswkJCXEuLyoqYvfu3dStWxd/f/9zntdms5GTk0NISIhrzmmqRHa7nYYNG/LQQw+RkpJyynWM6G/MmDHMmTOHjRs3nnadE495mTNnzgV/nqt7vNCfKVezWCzMnz+fPn364OPjY3Q5FcLTe1R/7u+8esw7DLMfgF2LHePmN8M1b4Bf1TvHsiK/h6f79/uftKdJKszhw4eZMWMGGRkZp703k4iIGGTPcvjqXsjLAO8A6P0KXHaPDsedgUKTVJioqCgiIyN5//33z/kQm4iIVBCbFZa9DktfBrsNIhs5DsdFNzG6sipPoUkqTFU+8jtmzJiznps0bdq0SqlFRKTS5GbCrCGwe6lj3OpO6PPaBd/w+GKh0CQiInIx+HMJfD0E8g857tfXdwK0ut3oqtyKQlMlqsp7XsS96GdJRMrNWgpLX4FlrwF2iGrqOBxX4xKjK3M7Ck2V4MRZ/gUFBae9uaPIuSgoKADw2KuARMRFcg7C1/fB3uWO8WUDHCd8++jfovOh0FQJvLy8CAsL49ChQwAEBgaWeS7Z2dhsNkpKSigqKnK7Ww6Uh6f3B67r0W63U1BQwKFDhwgLC8PLy8uFVYqIR9nxA8y+HwqOgm+Q487ezS/8UVYXM7cMTcuWLeO1115j3bp1HDx4kNmzZ9O/f//Trj9r1izeffddNm7cSHFxMU2bNmXMmDEkJSU51xkzZgxjx44ts12jRo3Ytm2bS2o+cZfpE8HpXNjtduejPM4lbLkLT+8PXN9jWFhYmTuXi4g4WS2w9EVY/oZjHNMcbpoGkQ0MLcsTuGVoys/Pp2XLlgwePJgbbrjhrOsvW7aMHj168NJLLxEWFsbUqVPp168fq1evpnXr/z2AsGnTpvzwww/Osbe36748JpOJmjVrEhUVhcViOadtLRYLy5Yt48orr/TIwzGe3h+4tkcfHx/tYRKRU/IvOYrXp9fBgTWOBe3ug54vgo/xN8H1BG4Zmnr37k3v3r3Lvf7EiRPLjF966SW++eYb5s6dWyY0eXt7V/j/3r28vM75HzwvLy9KS0vx9/f3yFDh6f3BxdGjiBjL9McCum57DrM1H/xC4Nq3oWl/o8vyKG4Zmi6UzWYjNzeXiIiIMst37NhBbGws/v7+dOzYkfHjx1O7du3TzlNcXExxcbFzfOLZaxaL5Zz3Jp3JiblcOWdV4un9gef36On9gef3qP7cmLUE80/P4736XccwugW2Gz+C8LrgQf1W5PewvHO6/bPnTCbTWc9p+qdXX32Vl19+mW3bthEVFQXA999/T15eHo0aNeLgwYOMHTuWtLQ0/vvf/xIcfOpn8JzqPCiA6dOnExgYeF79iIiIlFdA8WHa7ZlEeMGfAOyq0ZMtsbdiM2uP9rkoKCjgjjvuOOuz5y660DR9+nSGDBnCN998Q/fu3U+7XlZWFnXq1GHChAnce++9p1znVHua4uPjOXLkyBm/6OfKYrGQmppKjx49PPLQjqf3B57fo6f3B57fo/pzP6Zt8/Ca9wimomzs/qGU9H6DBXu8ParHv6vI72FOTg6RkZF6YO/fzZgxg/vuu48vv/zyjIEJHFcnXXLJJezcufO06/j5+eHn53fSch8fnwr5ga2oeasKT+8PPL9HT+8PPL9H9ecGSoth0UhY855jHNcW001TMAfFwp75ntHjGVREf+WdzzNvinMKn3/+OYMGDeLzzz+nb9++Z10/Ly+PXbt2UbNmzUqoTkREpByO/Qkf9fxfYOr0MAxeAOF1jK3rIuGWe5ry8vLK7AHavXs3GzduJCIigtq1azNixAjS0tL45JNPAMchuQEDBvDmm2+SmJhIRkYGAAEBAYSGhgLw5JNP0q9fP+rUqUN6ejqjR4/Gy8uL22/Xc3lERKQK2Dwbvn0EinMgIByufw8uSTr7duIybrmn6ddff6V169bO2wWkpKTQunVrRo0aBcDBgwfZt2+fc/3333+f0tJShg0bRs2aNZ2vRx991LnOgQMHuP3222nUqBG33HIL1atXZ9WqVdSoUaNymxMREfk7SxF8lwJfDnQEpvgOMHS5ApMB3HJPU5cuXc74wNJp06aVGS9ZsuSsc86YMeMCqxIREXGxIzsdYSnzd8e48+PQ9Tnwcst/vt2evuoiIiJV0aYv4bvHoCQPAiPhhvegwZkvYpKKpdAkIiJSlZQUwIKnYb3jvFzqdIYbP4QQXZhkNIUmERGRquLwdsfhuENbABNc+RRc9bQOx1UR+i6IiIhUBRunw7wnwFIA1aLgxg+gXhejq5K/UWgSERExUkk+zHsSfpvuGNe9Cm74AIKjja1LTqLQJCIiYpTMLfDlADjyB5jM0OVZuCIFzF5GVyanoNAkIiJS2ex2x4ne3w+H0iIIruk42Tuhs9GVyRkoNImIiFSm4lz47nH4/UvHuEF3x929q0UaW5eclUKTiIhIZTm4Cb4aBEd3gskLuo2ETo+C2S0f0HHRUWgSERGpaHY7/PoRLHgWrMUQEgc3TYHaHYyuTM6BQpOIiEhFKsqGuY86HrgLcEkv6P8uBEYYW5ecM4UmERGRipK+Ab4cBMd3g9kbuo+FjsPAZDK6MjkPCk0iIiKuZrfD6vdg0XNgs0Bobbh5KtRqa3RlcgEUmkRERFyp8Dh8kwzbvnOMG18D170DAeHG1iUXTKFJRETEVQ786jgcl70PvHyh5wvQ/n4djvMQCk0iIiIXymaDVZPghzFgK4XwBLh5GsS2NrgwcSWFJhERkQtRcAxmD4UdCx3jJv3h2rfAP9TQssT1FJpERETO175V8NVgyEkDLz/oNR7aDtbhOA+l0CQiInKubDb4ZSL8+ALYrRBR33E4rmYLoyuTCqTQJCIici7yDsPsB2DXYse4+c1wzRvgF2xsXVLhFJpERETKa89y+OpeyMsAb3/o8xq0vluH4y4SCk0iIiJnY7PCz/+GJePBboPIS+DmjyG6idGVSSVSaBIRETmT3EyYNQR2L3WMW94BfV8H32rG1iWVTqFJRETkdP5cAl8PgfxD4BMIff8Nre4wuioxiEKTiIjIP9mssORlWPYaYIeoJnDTVIhqbHRlYiCFJhERkb/LOQhf3wd7lzvGl90DvV4B30Bj6xLDKTSJiIicsPMHmHU/FBwF3yDHrQRa3GJ0VVJFKDSJiIhYS+GnF2D5G45xdHPHzSojGxhallQtCk0iInJxyz7guPfS/lWOcbv7oOeL4ONvbF1S5Sg0iYjIxeuPhY67exceB78Qx4N2m15vdFVSRSk0iYjIxcdqgcVjYcXbjnHNVnDzVIioZ2hZUrWZjS7gfCxbtox+/foRGxuLyWRizpw5Z91myZIlXHbZZfj5+dGgQQOmTZt20jqTJk0iISEBf39/EhMTWbNmjeuLFxERYx3fC1N7/y8wJT4I9y5SYJKzcsvQlJ+fT8uWLZk0aVK51t+9ezd9+/ala9eubNy4kccee4z77ruPhQsXOteZOXMmKSkpjB49mvXr19OyZUuSkpI4dOhQRbUhIiKVzLR9Prx3BRxYC/6hcOtn0Ptl8PYzujRxA255eK5379707t273OtPnjyZunXr8u9//xuASy+9lOXLl/PGG2+QlJQEwIQJExgyZAiDBg1ybjNv3jymTJnCM8884/omRESk8pQW0+zAp3hvWOQYx7WFm6ZAeB1j6xK34pah6VytXLmS7t27l1mWlJTEY489BkBJSQnr1q1jxIgRzvfNZjPdu3dn5cqVp523uLiY4uJi5zgnJwcAi8WCxWJxWf0n5nLlnFWJp/cHnt+jp/cHnt+jR/d3fDfmr++l/uFNAFgTH8LW9Tnw8gUP6tejv4dUbH/lnfOiCE0ZGRlER0eXWRYdHU1OTg6FhYUcP34cq9V6ynW2bdt22nnHjx/P2LFjT1q+aNEiAgNdf+fY1NRUl89ZlXh6f+D5PXp6f+D5PXpaf7HH19Bq30d42Qop8arG+jr3k1nSGhb+YHRpFcbTvof/VBH9FRQUlGu9iyI0VZQRI0aQkpLiHOfk5BAfH0/Pnj0JCQlx2edYLBZSU1Pp0aMHPj4+Lpu3qvD0/sDze/T0/sDze/S4/kqLMKeOxGvPVACsce34KewOruh7m2f0dwoe9z38m5JSG9n5RSxYvITb+rm+vxNHis7moghNMTExZGZmllmWmZlJSEgIAQEBeHl54eXldcp1YmJiTjuvn58ffn4nnzzo4+NTIT+wFTVvVeHp/YHn9+jp/YHn9+gR/R3ZCV8OhMzfHePOKdg6P0XRwlTP6O8sqmKPxaVWsgssZBVayC60kFVgIeev3+cUOX7NLSol569fc4st5BWVkldcSm5RKcWltr9m8ubO/t4u76+8810Uoaljx47Mnz+/zLLU1FQ6duwIgK+vL23atGHx4sX0798fAJvNxuLFi0lOTq7sckVE5Hz9/hXMfRRK8iCwOtzwPjTo7lHnLhnNarNzvKCEo3klHM0r5mh+Ccf+/iooIaughOP5FsevBRYKLVaXfLav2U6Bxcop9ldUCrcMTXl5eezcudM53r17Nxs3biQiIoLatWszYsQI0tLS+OSTTwAYOnQo77zzDsOHD2fw4MH8+OOPfPHFF8ybN885R0pKCgMGDKBt27a0b9+eiRMnkp+f77yaTkREqrCSAljwNKx3/L1Pncvhxg8hJNbYutxIfnEph3KLycwpIjOniEM5xRzOK+ZwruN15K/fHysowW4/9/nNJggN8CEs0JcQf29CAnwIDfAhJMCHEH8fQgK8Cfb3IcTfm2B/x++D/b0J8vMm2M8HX7ONRQsXEORnXHRxy9D066+/0rVrV+f4xHlFAwYMYNq0aRw8eJB9+/Y5369bty7z5s3j8ccf580336RWrVp8+OGHztsNANx6660cPnyYUaNGkZGRQatWrViwYMFJJ4eLiEgVc3i743DcoS2ACa58Cq56Grzc8p+4ClFYYiU9u5D0LMcrLauIjOxCDmYXkfHXK7e4tNzzmUwQHuhLRDVfqlfzpXqQL+GBjt+HV3P8PizQx/lrWKAvwX7emM2m8+6hKlwV6JY/UV26dMF+hph7qrt9d+nShQ0bNpxx3uTkZB2OExFxJxs/h3kpYCmAalGOw3H1u559Ow9TUmpj79F8th43cXz1Pg7mlHDgeCEHjhdw4HghR/NLyjVPoK8XMSH+RIX4ERXsT1SwHzX+9qpezY/IYF8iAn3x9nLL+2NfELcMTSIicpEryYd5T8Jv0x3julfCDR9CsOceHbBYbew/VsDuI/nO156j+ew9WkB6ViE2O4AXnOZWOdV8vYgLDyA2LICaoQHEhvoTE+pPzdAAYkL9iQ7xI9i/ap1AXtUoNImIiHvJ3OI4HHdkO5jMcNUzcOWTYPYyujKXKCyxsvNQHn9k5rLrcB67Duex81Aee48WUGo7/VEWfx8z4d5WmtSJonb1atQKDyQ+PIBa4YHEhQUQEuCNyXT+h8dEoUlERNyF3Q4b/gPzn4LSIgiKcZzsXfcKoys7L1abnd1H8tmWkcPWgzlsz8jlj8w89h8vOO2J1gE+XiREVqNeZDUSIgNJqF6NhMhq1IkIJMzfzPfff0+fPq2r3C0HPIVCk4iIVH3FufBdCvz+hWNc/2q4/n0IqmFsXeVUZLGy9WAOm9Nz2JyezeZ0R0j63/2HyqpezZeG0UE0iAqiQY0g6kcFUb9GEDVD/U+7t6gqnCjt6RSaRESkasv43XE47uhOMHnB1f+Cyx8Hc9U8Ebmk1Mb2jFx+O5DFpgNZbDqQzY5DeVhPcWgtwMeLRjHBXFozhMYxwTSMDuKS6GAigwy6EZGckUKTiIhUTXY7/DoFFowAazGExMGNH0GdjkZXVkZmThHr9x5n/b7jrN+Xxe9p2ZScYg9S9Wq+NIsLpWlsCE1jQ2kSG0KdiMALugxfKpdCk4iIVD1FOY47e2+e5Rg3TIL+70K16oaWZbfb2XkojzV7jrF29zHW7jlOWlbhSeuFBvjQolYoLWuF0aJWKM1rhRITcvpDa+IeFJpERKRqSd/oOBx3fDeYvaHbaOiYbMjhOLvdzo5DeazYeYSVfx5lze5jHC8oe+6Q2QSNY0K4rE4Yl9UOp3XtcBKqByogeSCFJhERqRrsdljzASz6F1hLILQ23DQF4ttVahlpWYUs33GYn3ccYdWfRzmSV/bGkP4+ZlrHh9OubgTtEyJoVTvM0Ed7SOXRd1lERIxXmAXfJsPWuY5x42vguncgILziP7rEyso/j7DsjyMs23GYPw/nl3nf38dM2zoRdKxfnQ71qtM8LhRf76p5ErpULIUmEREx1oF18NVAyNoHZh/o+QIkPuB4wFkF2Xe0gJ+2H+Kn7YdYuetomUv/zSZoFR9G54Y1uLx+dVrVDsPP2zNunCkXRqFJRESMYbfDqv+D1NFgs0B4Atw0FeIuc/lH2Wx29ubChNQdLN5+mD8y88q8HxcWwFWNanBlw0g61o8kNEA3h5STKTSJiEjlKzgGcx6CP753jJv0h2vfAv9Ql31EqdXG6t3HmP/7QVK3ZHIo1xvYDYCX2US7hHC6Noqia+MoGkYF6cRtOSuFJhERqVz7VsPX90L2fvDyg14vQdt7XXI4zmK1sWLXUb7//SALN2eUudLNz2zn6ktjSGpWk66NoggN1N4kOTcKTSIiUjlsNljxFiweB3YrRNSHm6dBzRYXOK2dtXuO8e1v6cz//WCZoBRRzZeeTaLp3jiS7D/Wcu01LfVcNjlvCk0iIlLx8o/A7KGwM9Uxbn4zXPMG+AWf95TbMnKYtT6Nub+lczC7yLk8MsiXXs1i6NOsJu3rRuDtZcZisTB/54U2IRc7hSYREalYe35xHI7LPQje/tD7VbjsnvM6HHckr5hvNqYza/0BNqfnOJcH+3vTq2kM17aKpWO96nh76ZYA4noKTSIiUjFsVvh5Aix5Cew2iLzEcTguuuk5TVNqtbH0j8PMWLufn7YdovSvB9/6eJm4unEU17euRZdGNfD30W0BpGIpNImIiOvlHYJZQ+DPJY5xy9uhz+vgF1TuKfYdLeCLX/fz5br9ZOYUO5e3ig/jxsviuKZFLOHVfF1cuMjpKTSJiIhr/bnUEZjyMsEnEPr+G1rdUa5NrTY7P207xH9W7WXpH4edyyOq+XLjZXHc0jaehtHnfx6UyIVQaBIREdewWWHpq7D0FcAONS51HI6LanzWTY/kFTNz7X6mr95HWlYh4DjlqXODSG5vX5vul0br0SViOIUmERG5cDkHHXuX9vzsGLe+23HCt2/gGTfbnJ7N1F/28O3GdEqsjkeZhAX6cGvbeO5IrE2d6tUqunKRclNoEhGRC7NzMcy6HwqOgG8QXDMRWtx82tWtNjs/bM1kyvLdrN59zLm8ZXwYAzrWoU/zmjqpW6okhSYRETk/1lLHlXE/TwDsEN3ccTgussEpVy+yWPlq3QE+/PlP9hwtAMDbbKJ385oMujyBy2qHV17tIudBoUlERM5ddprj3kv7VjrGbQdD0njw8T9p1eP5JXyyci+frNzD0fwSAEIDfLgzsTb3dEwgJvTkbUSqIoUmERE5N38sgtkPQOEx8A12PGi32Q0nrXYop4gPfv6Tz1bvo6DECkBcWAD3XVGXW9rGU81P/wSJe9FPrIiIlI/V4nhu3Iq3HOOaLR2H4yLqlVlt/7EC3lu2iy9+PUBJqePk7iY1QxjapT59msXobt3ithSaRETk7LL2wVeD4cBax7j9A9DzefD2c66y/1gBk37ayVfrDjjv2t22TjjDrm5Al0tqYDqPx6aIVCUKTSIicmbb5sGch6AoC/xC4bp3oMm1zrcPHHeEpS9//V9Y6twgkuSrG5BYN0JhSTyGQpOIiJxaaQn8MBpW/Z9jHNcGbpoC4QkAZOYU8faPO5i5dj8WqyMsXdEwkse6N6RNnQiDihapOApNIiJysuN74MtBkL7eMe6YDN1Gg7cvWQUlvLt0Fx+v2EORxXHO0uUNqvN490tom6CwJJ7Lbc/GmzRpEgkJCfj7+5OYmMiaNWtOu26XLl0wmUwnvfr27etcZ+DAgSe936tXr8poRUSkatnyDUy+0hGYAsLh9hmQ9CIFNjPv/LiDK179ifeW/kmRxUabOuHMuL8Dn93XQYFJPJ5b7mmaOXMmKSkpTJ48mcTERCZOnEhSUhLbt28nKirqpPVnzZpFSUmJc3z06FFatmzJzTeXvWNtr169mDp1qnPs5+eHiMjFwmwrwbzgaVj3kWNBfCLcNIXSoFi+WrOPCal/cCi3GIDGMcEM79WIro2idM6SXDTcMjRNmDCBIUOGMGjQIAAmT57MvHnzmDJlCs8888xJ60dElP3fz4wZMwgMDDwpNPn5+RETE1NxhYuIVFXH/uSKP57Hq3CvY9z5cexdnuWnncd5+fuf+SMzD4DaEYE80fMS+rWIxWxWWJKLi9uFppKSEtatW8eIESOcy8xmM927d2flypXlmuOjjz7itttuo1q1sg+CXLJkCVFRUYSHh3P11VfzwgsvUL169dPOU1xcTHFxsXOck5MDgMViwWKxnEtbZ3RiLlfOWZV4en/g+T16en/g2T2aNs/Ce34KYSV52AMisF73LlsC2zN+yjpW/ul4NlxYgA8PdanHHe3j8fM2Y7WWYrUaXPg58OTv3wme3mNF9lfeOU12u93u8k+vQOnp6cTFxbFixQo6duzoXD58+HCWLl3K6tWrz7j9mjVrSExMZPXq1bRv3965/MTep7p167Jr1y6effZZgoKCWLlyJV5ep35w5JgxYxg7duxJy6dPn05g4Jmf7C0iYjSzrYTmBz4j4ehPABwJasTSuAf56mAkqw6ZsGPC22Tnqpp2usfZCHS7/2aLlE9BQQF33HEH2dnZhISEnHa9i+6PwEcffUTz5s3LBCaA2267zfn75s2b06JFC+rXr8+SJUvo1q3bKecaMWIEKSkpznFOTg7x8fH07NnzjF/0c2WxWEhNTaVHjx74+Pi4bN6qwtP7A8/v0dP7Aw/s8egOvGfdh+noZuyYKO7wCC/+2ZrUbQHkFTt2IfVtFsOTPRtSKzzA4GIvnMd9/07B03usyP5OHCk6G7cLTZGRkXh5eZGZmVlmeWZm5lnPR8rPz2fGjBmMGzfurJ9Tr149IiMj2blz52lDk5+f3ylPFvfx8amQH9iKmreq8PT+wPN79PT+wEN6/G0mfPc4WPKhWhSbEl/jsdWh7D5aAFhpHhfKqH5NaOeBV8N5xPfvLDy9x4ror7zzud0tB3x9fWnTpg2LFy92LrPZbCxevLjM4bpT+fLLLykuLuauu+466+ccOHCAo0ePUrNmzQuuWUSkSijJhznDYPb9YMmnqFZnnqw+ievm+7D7aAHBPnZevr4p3wy73CMDk8iFcrs9TQApKSkMGDCAtm3b0r59eyZOnEh+fr7zarp77rmHuLg4xo8fX2a7jz76iP79+590cndeXh5jx47lxhtvJCYmhl27djF8+HAaNGhAUlJSpfUlIlJhDm2FLwfC4W3YTWZWxd/H4D+7UFhqwctsYkCH2jSy7OKGy+J0VZzIabhlaLr11ls5fPgwo0aNIiMjg1atWrFgwQKio6MB2LdvH2Zz2Z1o27dvZ/ny5SxatOik+by8vNi0aRMff/wxWVlZxMbG0rNnT55//nndq0lE3JvdDhs+hflPQWkhxQFRPG1/mDl/1AegU/3qjLm2KXUj/Jk/f5fBxYpUbW4ZmgCSk5NJTk4+5XtLliw5aVmjRo043YWCAQEBLFy40JXliYgYrzjPce7S718AsDWwHXcdG8xRQokO8WPkNU3o27wmJpPJYy9TF3Eltw1NIiJyBhm/Ow7HHd2JzeTFW7ZbePNYX0wmM4M6JZDS4xKC/T33ZGGRiqDQJCLiSex2WDcVvn8GrMUcNVfngcJh/GpvTMtaobx4fXOaxYUaXaWIW1JoEhHxFEU5MPdR2DwLgJ9srUkpegCLXwRjkxpxV4c6eOkkb5HzptAkIuIJ0jc6Dscd300pXrxiuZUPrX3odmlNnu/flJqh7n+DShGjKTSJiLgzux3WfIB90b8wWUtIs0eSXPIwB4Ka8X/XNqVXsxhMJu1dEnEFhSYREXdVmAXfPgxbv8UELLK24SnLAyS1bcy0Pk0IDdSJ3iKupNAkIuKODqzD9uVAzNn7KLF7Mb70DhYFXc/bN7bgyktqGF2diEdSaBIRcSd2O6z6P2ypozHbLOyz1SDZ8gitO1zNol6Nqeanv9ZFKor+dImIuIuCY5TOfhDvHQswA/Ot7Xkn6BFG3dyJDvWqn3VzEbkwCk0iIu5g/xqKP78Hv4KDFNu9eb70bkxt7+XLPpdq75JIJdGfNBGRqsxmw7L8Tbx+fB4/rOy2RTPWfzhD7u7P5Q0ija5O5KKi0CQiUlXlHyV3xr0E7/8JgG+snfi12Sjevq6tHoEiYgCFJhGRKsi6+xcKPx9IcMkhiuw+vO41mMSbU3i+aYzRpYlctBSaRESqEpuNrNSXCV75GkHY2GWryWe1xzLstv5UD/IzujqRi5pCk4hIFWHPzeTQxwOJPrICgG/sV2Lr+zoj21+iu3qLVAEKTSIiVUDe1h+xfnUv0dZjFNj9mBI6jOsGPkV8RKDRpYnIXxSaRESMZLOyf85YYje9jRc2/rDV4tf2b/Bgnx54mbV3SaQqUWgSETFIyfF0Dk69izo56wCY592d2ne9wx0JNQ2uTERORaFJRMQAGevn4T/3QerYs8m3+zE3fjjX3vMYgb76a1mkqtKfThGRSmS3Wtj2+bM02vEBZpOdbdQhM2kyt3XqZHRpInIWCk0iIpUk79BeMqbexaWFm8AEqYF9aDp4Eo0jI4wuTUTKQaFJRKQS7Foxi+qLHqEBueTaA/ilySh63PygTvYWcSMKTSIiFchmKWHTJ0/Qav8nAGwz1aP0xo/o1fwygysTkXN1QaHJYrGQkZFBQUEBNWrUICJCu5hFRE44mraLYx/fRauSLQAsCb2e1ve9Q2hwkMGVicj5MJ/rBrm5ubz77rtcddVVhISEkJCQwKWXXkqNGjWoU6cOQ4YMYe3atRVRq4iI29jy0wy8P7iShiVbyLEHsvyyN7jqsakKTCJu7JxC04QJE0hISGDq1Kl0796dOXPmsHHjRv744w9WrlzJ6NGjKS0tpWfPnvTq1YsdO3ZUVN0iIlWSpaSIXyc/QJOlDxBKHtu9GnL0rh/ofO1gPQpFxM2d0+G5tWvXsmzZMpo2bXrK99u3b8/gwYOZPHkyU6dO5eeff6Zhw4YuKVREpKrL2LuN3P/cTdvSPwBYXuNW2t77Jv7+AQZXJiKucE6h6fPPPy/Xen5+fgwdOvS8ChIRcUcbF35CvZVPE0MB2fZq/NHxNTr3utPoskTEhc77RHCr1cqHH37I9u3bqVWrFi1btqRVq1ZUr17dlfWJiFRpxUUFbPzoYRIPfwXANu/GBN/1H9olXGJwZSLiaucdmh5++GG+/vprunfvzjvvvIPJZKK0tJS4uDhatWrFt99+68o6RUSqnAM7/0vR5/eQaN0FwIqYu2g7aAK+fn4GVyYiFeGcr547YdasWXzyySd89tln+Pn58euvv/Lmm29SVFREnTp1XFmjiEiVs27+h4T9pzsNrLs4TjAbr3ifTkMnKTCJeLDzDk15eXk0adIEAB8fH7y9vUlOTmbEiBH4+/u7rMDTmTRpEgkJCfj7+5OYmMiaNWtOu+60adMwmUxlXv+s0W63M2rUKGrWrElAQADdu3fX1X8icpKigjxWv30PbdY8QZCpkC0+zSgZsoxW3W41ujQRqWDnHZrq1atHeno6AHFxcaSlpQHQr18/Pv30U9dUdxozZ84kJSWF0aNHs379elq2bElSUhKHDh067TYhISEcPHjQ+dq7d2+Z91999VXeeustJk+ezOrVq6lWrRpJSUkUFRVVaC8i4j6Ksg5y+M0uJB79BpvdxMq4wVwy/Cei4+oZXZqIVILzDk033HAD33//PQBXXXUVU6ZMAWDLli0UFha6prrTmDBhAkOGDGHQoEE0adKEyZMnExgY6KzhVEwmEzExMc5XdHS08z273c7EiRN57rnnuO6662jRogWffPIJ6enpzJkzp0J7ERH3sOG797nmz1HUs+3hKKFs7jaVjkPewNvH1+jSRKSSnPeJ4GPGjHH+fvjw4bRr144aNWqQk5PDvffe64raTqmkpIR169YxYsQI5zKz2Uz37t1ZuXLlabfLy8ujTp062Gw2LrvsMl566SXn/aZ2795NRkYG3bt3d64fGhpKYmIiK1eu5LbbbjvlnMXFxRQXFzvHOTk5gOPxMhaL5YL6/LsTc7lyzqrE0/sDz+/Rk/srzM9l67RhJGbNBxP87tuSyLun0jimtkf168nfQ/D8/sDze6zI/so7p8lut9td8YFHjhxh7ty5VK9enX79+lXYnW/T09OJi4tjxYoVdOzY0bl8+PDhLF26lNWrV5+0zcqVK9mxYwctWrQgOzub119/nWXLlrF582Zq1arFihUruPzyy0lPT6dmzZrO7W655RZMJhMzZ848ZS1jxoxh7NixJy2fPn06gYGBLuhWRIxUdDyNtnsmUZ8DWO0m5le7ntKG12I2n/dOehGpggoKCrjjjjvIzs4mJCTktOtd0AN7/y4yMpJBgwa5ajqX6tixY5mA1alTJy699FLee+89nn/++fOed8SIEaSkpDjHOTk5xMfH07NnzzN+0c+VxWIhNTWVHj164OPj47J5qwpP7w88v0dP689us7H+u3dpuftFAkwlHCactK5vYMs2k+QhPf6Tp30P/8nT+wPP77Ei+ztxpOhszik07du3j9q1a5d7/bS0NOLi4s7lI84qMjISLy8vMjMzyyzPzMwkJiamXHP4+PjQunVrdu7cCeDcLjMzs8yepszMTFq1anXaefz8/PA7xeXFPj4+FfIDW1HzVhWe3h94fo+e0F9+bhZbPhxCh+xFYIJN/m2IG/QJTSOi2Tt/vkf0eCbqz/15eo8V0V955zunfczt2rXjgQceYO3ataddJzs7mw8++IBmzZrx9ddfn8v05eLr60ubNm1YvHixc5nNZmPx4sVl9iadidVq5ffff3cGpLp16xITE1NmzpycHFavXl3uOUXE/e36fTVHJlxOu+xFlNrNrKw7jGZPpVI9upbRpYlIFXBOe5q2bNnCiy++SI8ePfD396dNmzbExsbi7+/P8ePH2bJlC5s3b+ayyy7j1VdfpU+fPhVSdEpKCgMGDKBt27a0b9+eiRMnkp+f7zw8eM899xAXF8f48eMBGDduHB06dKBBgwZkZWXx2muvsXfvXu677z7AcWXdY489xgsvvEDDhg2pW7cuI0eOJDY2lv79+1dIDyJSddhtNlZ//Qat/jsef5OFQ0RwtM9kOiYmGV2aiFQh5xSaqlevzoQJE3jxxRf56quvWLduHXv37qWwsJDIyEjuvPNOkpKSaNasWUXVC8Ctt97K4cOHGTVqFBkZGbRq1YoFCxY4byOwb9++MidqHj9+nCFDhpCRkUF4eDht2rRhxYoVzptzguNE8vz8fO6//36ysrLo3LkzCxYsqJQbdYqIcbKzjvHHh4PpkPcTmOC3gPbUHvwJl9aoefaNReSicl4nggcEBDBw4EBmzpzJxIkTXVxS+SQnJ5OcnHzK95YsWVJm/MYbb/DGG2+ccT6TycS4ceMYN26cq0oUkSpu+4blBH57H+3sB7HYvVjf8BHa3zESk9nL6NJEpAo676vn7HY777//Pm+88QYmk4m2bdty55130q5dO1fWJyLicjarjZUzX6Hd9tfxNZWSYapBzjXvkdi2m9GliUgVdkE3G9mwYQOXXXYZnTt3ZvPmzVxxxRU8+eSTrqpNRMTljh05xLp/X8vlf7yMr6mUTdU6EfjICi5RYBKRs7ig+zRNnz6dHj16OMebNm3iuuuuIy4ujscff/yCixMRcaVNq3+k+vcP0I5DlNi92HRpCm1ueRaTblYpIuVw3n9TREREEB8fX2ZZixYteOedd3j33XcvuDAREVcpLbXy07QxNJ5/E3Ec4qApmvQb59D2tucUmESk3M77b4tWrVoxderUk5Y3aNCAffv2XVBRIiKuknYwjQ2v9aHrnjfwNVn5PeQqwh5fRUKLK40uTUTczHkfnnvhhRfo2rUr6enpPPTQQ7Ro0YL8/Hxeeukl6tat68oaRUTOyy8/zafu0odpxxFK7N5sa/UMLfo/CRX0bEwR8WznHZo6dOjAqlWrePTRR7niiis48dxff39/vvzyS5cVKCJyrgqKS1gydRQ9Dr6Pj8nKQa+amG6eRovGHYwuTUTc2AWdCN6yZUuWLFnCoUOHWLduHTabjcTERCIjI11Vn4jIOdn2526yPruXPtZ1YIKt1bvT4N6P8AkMM7o0EXFzFxSaToiKiqJ3796umEpE5LzYbHbmffc17dY9RWPTMYrxYX/iaC7tlazDcSLiEi4JTSIiRjqUXcCSKc9yQ9bHeJtsZPjUIuCO/9Cg7mVGlyYiHkShSUTc2pL1W/D7dii38BuY4M/Ya6g7YDImv2CjSxMRD6PQJCJuKb+4lOkzP+W6XaOJMmVRhB/ZXV+k3pX36XCciFQIhSYRcTsb9hxh42f/YnDJTLxMdg4H1CX07v8QHdvc6NJExIMpNImI2yi12pi6cDVNVz3BIPNmMEFm/ZuIvvUt8K1mdHki4uEUmkTELew6nMfHn07j4axXqGHOodjkj7XPBKLb3Wl0aSJykVBoEpEqzWaz8+mKXeQveoExpjmYTXayQy4h9O7PoMYlRpcnIhcRhSYRqbIOZhfy0owfuSttHInmbQDkN7+H0GtfBZ8Ag6sTkYuNQpOIVDl2u52v1h3gx7mf8QLvUN2cS4lXNbyve4tqLW4yujwRuUgpNIlIlZKZU8RzX2+gza5JvOs9F4DiyGb43f4JVK9vcHUicjFTaBKRKsFutzNnYxrvfrOU8bY3aOO9AwBb2/vwS3oRfPwNrlBELnYKTSJiuIzsIv41+3dMf3zPFz6TCTPnY/UNwav/O5ibXGd0eSIigEKTiBjIbrczc+1+Xpn3O8Os/+E+3+8dy2Mvw+umKRBR1+AKRUT+R6FJRAyx/1gBz8zaxN5dW5nq8xatvP90vNHhIUzdx4K3r7EFioj8g0KTiFSqUquNKb/sZkLqH1xlXcV83/cJMRVg9w/D1P9daNzH6BJFRE5JoUlEKs1/07J5+utN7Eg/yrPenzHQd5HjjVrtMN00BcJqG1ugiMgZKDSJSIUrKCnljdQ/+Gj5buLJYI7/2zRht+PNTo9At1Hg5WNskSIiZ6HQJCIVatHmDMbO3UJaViHXmFfymv9HBNgKICACrp8MlyQZXaKISLkoNIlIhThwvIAx327hh62Z+FHCxGrT6W9dBDagdke48SMIjTO6TBGRclNoEhGXKim18dHy3by1eAeFFisNzQf5LOxdogp2Aia4IgW6PAte+utHRNyL/tYSEZf5eecRXpi3nT+P5APwRPQGhuVPwlxQAIGRcMP70KCbwVWKiJwfhSYRuWAHjhfy0XYzm1auByCuGnwa+xV1989yrJBwBdzwAYTUNLBKEZELYza6gPM1adIkEhIS8Pf3JzExkTVr1px23Q8++IArrriC8PBwwsPD6d69+0nrDxw4EJPJVObVq1evim5DxK0VlJQyIfUPer31C5uOmfEym3i6DSwLH/dXYDLBVc/APd8oMImI23PLPU0zZ84kJSWFyZMnk5iYyMSJE0lKSmL79u1ERUWdtP6SJUu4/fbb6dSpE/7+/rzyyiv07NmTzZs3Exf3vxNRe/XqxdSpU51jPz+/SulHxN3YbHa++S2NV77fTkZOEQANQmx83OEgcStGQWkhVIuCGz+EelcZXK2IiGu4ZWiaMGECQ4YMYdCgQQBMnjyZefPmMWXKFJ555pmT1v/ss8/KjD/88EO+/vprFi9ezD333ONc7ufnR0xMTMUWL+Lm1u09zvPfbWHj/iwAaoUH8Gy3OJr98jhxy35xrFSvi+NwXNDJ/4kREXFXbheaSkpKWLduHSNGjHAuM5vNdO/enZUrV5ZrjoKCAiwWCxEREWWWL1myhKioKMLDw7n66qt54YUXqF69+mnnKS4upri42DnOyckBwGKxYLFYzqWtMzoxlyvnrEo8vT/wjB73Hi3g9dQdLNicCUA1Xy8evKoegxvk4f/NHZiP78RuMmO78hlsnR4Fsxe4cb//5AnfwzNRf+7P03usyP7KO6fJbrfbXf7pFSg9PZ24uDhWrFhBx44dncuHDx/O0qVLWb169VnneOihh1i4cCGbN2/G398fgBkzZhAYGEjdunXZtWsXzz77LEFBQaxcuRIvL69TzjNmzBjGjh170vLp06cTGBh4nh2KVC15Flh4wMwvmSasdhMm7LSvYadvvJUWuUtofuBTvOwWCn3CWZfwIEeDGhtdsojIOSkoKOCOO+4gOzubkJCQ067ndnuaLtTLL7/MjBkzWLJkiTMwAdx2223O3zdv3pwWLVpQv359lixZQrdup75EesSIEaSkpDjHOTk5xMfH07NnzzN+0c+VxWIhNTWVHj164OPjeY+a8PT+wD17zC8u5eOV+/hg+R7yiksBuLJhdYb3vIRG4eA1PwXz/tkAWOt2ZUm1G7mqz41u09+5csfv4blQf+7P03usyP5OHCk6G7cLTZGRkXh5eZGZmVlmeWZm5lnPR3r99dd5+eWX+eGHH2jRosUZ161Xrx6RkZHs3LnztKHJz8/vlCeL+/j4VMgPbEXNW1V4en/gHj0Wl1r5fPU+3vlpJ0fySgBoUjOEZ/tcSueGkXDwN5gyEI79CSYv6DYKW/sHKfl+gVv0d6E8vUf15/48vceK6K+887ldaPL19aVNmzYsXryY/v37A2Cz2Vi8eDHJycmn3e7VV1/lxRdfZOHChbRt2/asn3PgwAGOHj1KzZq6TFouDqVWG7M3pDHxhx2kZRUCUKd6II93v4RrW8ZiNgFrPoCFz4K1BEJqwU1ToHaiR527JCJyOm4XmgBSUlIYMGAAbdu2pX379kycOJH8/Hzn1XT33HMPcXFxjB8/HoBXXnmFUaNGMX36dBISEsjIyAAgKCiIoKAg8vLyGDt2LDfeeCMxMTHs2rWL4cOH06BBA5KS9DBR8WylVhvf/pbO2z/uZPdfd/KOCvbjkW4NubVdPD5eZijKhm8fhi3fODa6pDf0/z8IjDjDzCIinsUtQ9Ott97K4cOHGTVqFBkZGbRq1YoFCxYQHR0NwL59+zCb/3ffznfffZeSkhJuuummMvOMHj2aMWPG4OXlxaZNm/j444/JysoiNjaWnj178vzzz+teTeKxrDY7c39L563FO5yPPYmo5ssDV9bjno4JBPj+dQFE2nr4ciBk7QWzD/QYCx0eApPJuOJFRAzglqEJIDk5+bSH45YsWVJmvGfPnjPOFRAQwMKFC11UmUjVVlJqY86GNN5dusu5Zyks0If7r6zHgI4JVPP7668Fux1WT4ZFI8FmgbDacNM0qNXGuOJFRAzktqFJRM5NkcXKzLX7eW/pLtKzHXfxDgv0YcgV9RjQKYEgv7/9dVBwDL5Jhu3zHONL+8G170BAWOUXLiJSRSg0iXi4rIISPl21l2kr9jivhqsR7MeQK+pyR2KdsmEJYP9a+GoQZO8HL1/o+SK0H6LDcSJy0VNoEvFQ+48V8NHy3Xzx634KSqwAxIUFMPSqetzcNh5/n3/ctNVmg5Vvw+JxYCuF8Lpw8zSIbVXptYuIVEUKTSIexG63s37fcab8socF/83AanPc8P/SmiHcf2VdrmkR67ga7p/yj8KcB2HHX+f2Nb0B+r0J/q67SauIiLtTaBLxACWlNub/fpApv+xm04Fs5/IrGkZy/5X16NwgEtPpDq/tXQlfDYbcdPDyg94vQ5tBOhwnIvIPCk0ibuxgdiGfr9nPjDX7OJTreHi0r7eZ/q1iGdipLk1iz7CnyGaDX96AH18EuxWqN3AcjotpXjnFi4i4GYUmETdjs9lZseso/1m1hx+2HnIegosK9uPuDnW4I7E21YPOcn+xvMMw+37Y9aNj3OJW6DsB/IIquHoREfel0CTiJg7lFPHV+gN8sXY/e44WOJe3rxvB3R3qkNQ0Bl/vU5yv9E+7f4av74O8DPAOgD6vQeu7dDhOROQsFJpEqjCL1cbS7YeZ+et+ftz2v71KQX7e3HBZHHd1qMMl0cHlm8xmhWWvw9KXwW6DyEZwy8cQdWkFdiAi4jkUmkSqGLvdzpaDOcxan8Y3G9Oc91YCaFMnnFvbxdO3ec3/3bm7PHIzYdZ9sHuZY9zqLujzKvhWc3H1IiKeS6FJpIpIzypk7m/pzN6QxraMXOfy6tV8ueGyOG5tF0+DqHLuVfq7XT/BrCGQfxh8qsE1E6DlbS6sXETk4qDQJGKgo3nFzP9vBt9uTGPtnuPO5b5eZno0ieaGy+K48pIap7630tlYSx2H4pa9Dtghqqnj6rgal7isfhGRi4lCk0glO5JXzMLNGSz4bwYrdh11nqdkMkG7hAiuaxXLNc1jCQ30Of8PyUl3nOy99xfHuM1A6PUy+ARceAMiIhcphSaRSpCeVciygyamT1nL2j3H+SsnAdA8LpRrW8ZyTcua1Ax1QajZ8YPjdgIFR8E3yHFn7+Y3Xfi8IiIXOYUmkQpgt9vZnJ5D6pZMftiayeb0HMALcByCa1ErlN7NatK7WQwJkS46GdtqgR9fgF8mOsYxLRyH46rXd838IiIXOYUmERfJKy7ll51HWLL9ED9tO0xGTpHzPZMJ6gbZublTI65pGUd8RKBrPzz7gONRKPtXO8bt7oOeL4KPv2s/R0TkIqbQJHKebDbHrQGW7zzCzzsOs2b3MSzW/x138/cxc0XDGvRoEs2VDSJYvfQH+nROwMfnAs5VOpXtC2DOUCg8Dn4hcO3b0LS/az9DREQUmkTOxf5jBazYdYRfdh7ll51HOJpfUub9OtUD6dooiq6No0isG4G/jxcAFovF9cWUlsDisbDyHcc4tjXcNBUi6rr+s0RERKFJ5HTsdjsHjheyZvcxVv55lJW7jpKWVVhmnWq+XnSoV50rGkZy5SU1qFejkp7ddnwvfDUI0tY5xh0egu5jwdu3cj5fROQipNAk8herzc4fmbn8uucYa/YcZ+3uY2XOSwLwNptoGR9Gp/rV6dwgkta1w8v3vDdX2joXvhkGRdngHwr934XGfSu3BhGRi5BCk1y0juYVs+lANuv3HWf9vuP8tj+bvOLSMut4m000iwulQ73qdKxfnbZ1ws/t8SWuVFoMi0bCmvcc41rt4KYpEFbbmHpERC4yCk1yUcgutLAlPYff07L4bX82vx3I4sDxwpPWC/LzplV8GO0SImhXN5zW8eEE+HoZUPE/HPsTvhwEBzc6xp0ehm6jwcvFJ5WLiMhpKTSJR7Hb7RzMLmJbRg5bD+by37RsNqfnsO9YwSnXr1ejGq3jw2lTJ5zL6oTRMCoYL7Opkqs+i82z4dtHoDgHAiLg+slwSZLRVYmIXHQUmsRtZRdY+ONQLn9k5rIjM4+tB3PYlpFLduGpr1SrFR5A87hQWtQKo2WtUJrVCiXEvwrvqbEUwcJn4dePHOPaHeHGjyA0zti6REQuUgpNUqXZbHYyc4vYdSifXYfznK8dmXkcyi0+5TZeZhP1a1SjcUwIzeJCaBYbSpPYEMIC3ejKsiM74cuBkPm7Y9w5Bbr+C7z0R1ZExCj6G1gMZ7PZOV4Mq3cfIy27mL1HC9h9JJ/dR/LZczSfIovttNvGhvpzSUwwl0Q7XpfWDKZBVBB+3lXgPKTztekLmPsYWPIhMBJueA8adDe6KhGRi55Ck1Q4m83Okbxi0rIKScsqZP+xQg4cL+DA8UL2Hy9g/7ECLFZvWP/rKbf3MpuoUz2Q+jWC/npVo35UEA2jggiuyofXzlVJAXw/HDb8xzFOuAJu+ABCahpbl4iIAApNcoFKrTaO5JWQmVNERk4RmTlFHMwuIiO7iIPZhaRnOX79++NFTsVsshMfXo06kdWoExFIQmQ16kVWIyGyGrXCA/DxquR7IVW2Q9sch+MObwVMcNVwuOppMLvxHjMREQ+j0CQnsdnsZBdaOJpfzJG8Eo7kFXMkt5jDecUczv3rlVdMZk4xR/KKsZ85DwFgNkFMiD+xYQHERwRSKzzgr1cgNUN82LhiCf36dnb9c9ncwYbPYP6TYCmAalFw4wdQr4vRVYmIyD8oNHm4UquNnKJSsgpKyC60kFVoIaughGP5jl+PF5RwPN8RkBy/OpZZbeVIQn/xMpuoEeRHdIgfMaH+1AwNICbU3xmS4sIDiA72w/s0e4ssFgu/V7Gr/CtFST7MewJ++9wxrtfFcTguKMrQskRE5NQUmqq4E/cdOlgAG/ZlUWiFvKJScoss5P71a05RqfP32YWOcU6hhZxCC7n/uMP1uQjx9yYy2I/Ian5UD/IlKtiPGn+9IoP8iA7xJyrEj+rV/KrevY2quszNjsNxR/4Akxm6PAtXpOhwnIhIFea2oWnSpEm89tprZGRk0LJlS95++23at29/2vW//PJLRo4cyZ49e2jYsCGvvPIKffr0cb5vt9sZPXo0H3zwAVlZWVx++eW8++67NGzYsDLaOaOr/r0Mu90bfltz3nME+3kTEuBDWKAP4YG+zl/DA30Ir+ZLxD9e1av5Vf4z1S4GdjusmwbfPw2lRRBcE278EBI6G12ZiIichVuGppkzZ5KSksLkyZNJTExk4sSJJCUlsX37dqKiTj60sWLFCm6//XbGjx/PNddcw/Tp0+nfvz/r16+nWbNmALz66qu89dZbfPzxx9StW5eRI0eSlJTEli1b8Pf3r+wWnUwmE+GBPhQXlxARHEiwvw9B/t4E+XkT7O9NiL8Pwf7eBP/1a2iADyEBPo5f/zb2+BOp3YC3tRCvbx6AzbMcCxp0h+vfg2qRxhYmIiLl4pahacKECQwZMoRBgwYBMHnyZObNm8eUKVN45plnTlr/zTffpFevXjz11FMAPP/886SmpvLOO+8wefJk7HY7EydO5LnnnuO6664D4JNPPiE6Opo5c+Zw2223VV5zp7D6ma7Mnz+fPn2uuDhPlPYEGb9z1fZRmIszweQF3UZCp0fBrDArIuIu3C40lZSUsG7dOkaMGOFcZjab6d69OytXrjzlNitXriQlJaXMsqSkJObMmQPA7t27ycjIoHv3/91AMDQ0lMTERFauXHna0FRcXExx8f/uSp2TkwM4Tmy2WE79KI/zcWIuV85ZlXh0f3Y75nVT8f5hJD7WYmzBsdiu/wB7fCJYrY6XB/Do7+FfPL1H9ef+PL3HiuyvvHO6XWg6cuQIVquV6OjoMsujo6PZtm3bKbfJyMg45foZGRnO908sO906pzJ+/HjGjh170vJFixYRGBh49mbOUWpqqsvnrEo8rT9vawGt9n1EXNZaADJCWrG+zhAsvx+F3+cbXF3F8LTv4al4eo/qz/15eo8V0V9Bwakf6v5PbheaqpIRI0aU2YOVk5NDfHw8PXv2JCQkxGWfY7FYSE1NpUePHh55eM4T+zOlb8Br9hBMWXuwm72xXPUvVh+vR4+ePT2mx7/zxO/hP3l6j+rP/Xl6jxXZ34kjRWfjdqEpMjISLy8vMjMzyyzPzMwkJibmlNvExMSccf0Tv2ZmZlKzZs0y67Rq1eq0tfj5+eHn53fSch8fnwr5ga2oeasKj+jPbofVk2HRSLBZILQ2ppunYopuCfPne0aPZ+Dp/YHn96j+3J+n91gR/ZV3Prc7C9XX15c2bdqwePFi5zKbzcbixYvp2LHjKbfp2LFjmfXBsXvvxPp169YlJiamzDo5OTmsXr36tHOKnKTwOMy8CxY84whMja+BocugVlujKxMRERdwuz1NACkpKQwYMIC2bdvSvn17Jk6cSH5+vvNqunvuuYe4uDjGjx8PwKOPPspVV13Fv//9b/r27cuMGTP49ddfef/99wHHZf2PPfYYL7zwAg0bNnTeciA2Npb+/fsb1aa4kwO/wpeDIHsfePlCzxeg/f1g0k0/RUQ8hVuGpltvvZXDhw8zatQoMjIyaNWqFQsWLHCeyL1v3z7Mf7uUu1OnTkyfPp3nnnuOZ599loYNGzJnzhznPZoAhg8fTn5+Pvfffz9ZWVl07tyZBQsWGHqPJnEDNhusmgQ/jAFbKYQnwM3TILa1wYWJiIiruWVoAkhOTiY5OfmU7y1ZsuSkZTfffDM333zzaeczmUyMGzeOcePGuapE8XQFx2D2UNix0DFuej30exP8Q42tS0REKoTbhiYRQ+1bBV8Nhpw08PKD3i9Dm0E6HCci4sEUmkTOhc0Gv0yEH18AuxWqN3AcjotpbnRlIiJSwRSaRMor7zDMfgB2/XWVZfNb4JoJ4BdsbF0iIlIpFJpEymPPcvjqXsjLAO8A6PMqtL5bh+NERC4iCk0iZ2KzwrLXYenLYLdBZCPH4bjoJkZXJiIilUyhSeR0cjNh1n2we5lj3OpO6PMa+FYzti4RETGEQpPIqez6CWbdD/mHwCcQ+k6AVrcbXZWIiBhIoUnk76yljkNxy14H7BDVBG7+GGpcYnRlIiJiMIUmkRNy0uHrIbB3uWN82QDo/Qr4BBhbl4iIVAkKTSIAO36A2fdDwVHwDXLc2bv5TUZXJSIiVYhCk1zcrBb46UVY/oZjHNMcbpoGkQ0MLUtERKoehSa5eGUfcDwKZf9qx7jdfdDzRfDRQ5pFRORkCk1ycdq+AOYMhcLj4BcC177leOCuiIjIaSg0ycWltAQWj4WV7zjGNVvBzVMhop6hZYmISNWn0CQXj+N7HYfj0n51jBMfhB5jwdvP2LpERMQtKDTJxWHrd/DNQ1CUDf6hcN3/waXXGF2ViIi4EYUm8WylxZA6ClZPdozj2sJNUyC8jrF1iYiI21FoEs917E/4chAc3OgYd3oYuo0GLx9DyxIREfek0CSeafNs+PYRKM6BgHC4/j24JMnoqkRExI0pNIlnsRTBwmfh148c4/gOcNNHEFrL2LpERMTtKTSJ5ziyE74cCJm/O8adU6Drv8BLP+YiInLh9K+JeIZNX8J3j0FJHgRGwg3vQYPuRlclIiIeRKFJ3FtJASx4GtZ/4hgnXAE3fAAhNY2tS0REPI5Ck7ivw9sdh+MObQFMcNVwuOppMHsZXZmIiHgghSZxTxunw7wnwFIA1aLgxg+gXhejqxIREQ+m0CTupSQf5j0Jv013jOte5TgcFxxtbF0iIuLxFJrEfWRugS8HwJE/wGSGLiPgiid0OE5ERCqFQpNUfXa740Tv74dDaREE14QbP4SEzkZXJiIiFxGFJqnainPhu8fh9y8d4wbdHXf3rhZpbF0iInLRUWiSquvgJsfVccd2gckLuo2ETo+C2Wx0ZSIichFSaJKqx253PAZlwbNgLYaQOLhpCtTuYHRlIiJyEXO7/7IfO3aMO++8k5CQEMLCwrj33nvJy8s74/oPP/wwjRo1IiAggNq1a/PII4+QnZ1dZj2TyXTSa8aMGRXdjvxTUbZj79K8JxyB6ZJeMHS5ApOIiBjO7fY03XnnnRw8eJDU1FQsFguDBg3i/vvvZ/r06adcPz09nfT0dF5//XWaNGnC3r17GTp0KOnp6Xz11Vdl1p06dSq9evVyjsPCwiqyFfkHU/oGmDMEju8Bszd0Hwsdh4HJZHRpIiIi7hWatm7dyoIFC1i7di1t27YF4O2336ZPnz68/vrrxMbGnrRNs2bN+Prrr53j+vXr8+KLL3LXXXdRWlqKt/f/vgRhYWHExMRUfCNSlt1OvUOL8Pp4JtgsEFobbp4KtdoaXZmIiIiTW4WmlStXEhYW5gxMAN27d8dsNrN69Wquv/76cs2TnZ1NSEhImcAEMGzYMO677z7q1avH0KFDGTRoEKYz7OUoLi6muLjYOc7JyQHAYrFgsVjOpbUzOjGXK+esMgqzMM1NpnnaAgBsl/TBes1bEBAGHtSvR38P8fz+wPN7VH/uz9N7rMj+yjunW4WmjIwMoqKiyizz9vYmIiKCjIyMcs1x5MgRnn/+ee6///4yy8eNG8fVV19NYGAgixYt4qGHHiIvL49HHnnktHONHz+esWPHnrR80aJFBAYGlquec5GamuryOY0Unr+LtnsmEVhyBJvJi82xt/NnYA/4aYXRpVUYT/se/pOn9wee36P6c3+e3mNF9FdQUFCu9apEaHrmmWd45ZVXzrjO1q1bL/hzcnJy6Nu3L02aNGHMmDFl3hs5cqTz961btyY/P5/XXnvtjKFpxIgRpKSklJk/Pj6enj17EhIScsH1nmCxWEhNTaVHjx74+Pi4bF7D2O2YV/8f5t9exGQrxRZWh5+jB9Huugdo7An9nYLHfQ//wdP7A8/vUf25P0/vsSL7O3Gk6GyqRGh64oknGDhw4BnXqVevHjExMRw6dKjM8tLSUo4dO3bWc5Fyc3Pp1asXwcHBzJ49+6xf8MTERJ5//nmKi4vx8/M75Tp+fn6nfM/Hx6dCfmArat5KVXAM5jwIfzgOx9HkOqy93yDrx+We0d9ZeHqPnt4feH6P6s/9eXqPFdFfeeerEqGpRo0a1KhR46zrdezYkaysLNatW0ebNm0A+PHHH7HZbCQmJp52u5ycHJKSkvDz8+Pbb7/F39//rJ+1ceNGwsPDTxuY5DzsWwVfDYacNPDyg14vQdt7obTU6MpERETOqkqEpvK69NJL6dWrF0OGDGHy5MlYLBaSk5O57bbbnFfOpaWl0a1bNz755BPat29PTk4OPXv2pKCggE8//ZScnBznbrgaNWrg5eXF3LlzyczMpEOHDvj7+5OamspLL73Ek08+aWS7nsNmg18mwo8vgN0KEfXh5mlQs4XRlYmIiJSbW4UmgM8++4zk5GS6deuG2Wzmxhtv5K233nK+b7FY2L59u/OkrvXr17N69WoAGjRoUGau3bt3k5CQgI+PD5MmTeLxxx/HbrfToEEDJkyYwJAhQyqvMU+VfwRmPwA7f3CMm90E/SaCX7ChZYmIiJwrtwtNERERp72RJUBCQgJ2u9057tKlS5nxqfTq1avMTS3FRfYsh6/vg9yD4O0PfV6D1nfrZpUiIuKW3C40iRuwWeHnCbDkJbDbILKR43BcdBOjKxMRETlvCk3iWnmHHHuXdi91jFveAX1fB99qxtYlIiJygRSaxHX+XAJfD4H8Q+ATCH0nQKvbja5KRETEJRSa5MLZrLD0FVj6KmCHqCaOw3E1GhldmYiIiMsoNMmFyTnoOBy3d7ljfNkA6P0K+AQYW5eIiIiLKTTJ+dv5A8y6HwqOgm8QXDMRWtxsdFUiIiIVQqFJzp21FH56AZa/4RhHN3ccjotscMbNRERE3JlCk5yb7APw1b2wf5Vj3O4+6Pki+Jz90TQiIiLuTKFJym/7ApgzFAqPg18IXPsWNL3e6KpEREQqhUKTnF1pCSweCyvfcYxrtoKbp0JEPUPLEhERqUwKTXJmx/fCV4Mh7VfHOHEo9BgH3n7G1iUiIlLJFJrk9LZ+B988BEXZ4B8K102CS/sZXZWIiIghFJrkZKXFkDoKVk92jOPawE1TIbyOsXWJiIgYSKFJyjq2G74cCAc3OsYdk6HbaPD2NbIqERERwyk0yf9sng3fPgLFORAQDv3fhUa9ja5KRESkSlBoErAUwcJn4dePHOP4RLjxIwiLN7YuERGRKkSh6WJ3dBd8OQAyfneMOz8OXf8FXj7G1iUiIlLFKDRdzH7/CuY+CiV5EFgdrn8fGnY3uioREZEqSaHpYmQphO+Hw/pPHOM6l8ONH0JIrLF1iYiIVGEKTRebw9sdV8cd2gKY4Mon4apnwEs/CiIiImeifykvJhs/h3kpYCmAalFww/tQv6vRVYmIiLgFhaaLQUk+zH8KNn7mGNe9Em74EIKjja1LRETEjSg0ebrMLY7DcUe2g8nsOBR35ZNg9jK6MhEREbei0OSp7HbY8B+YPxxKCyEoxnGyd90rjK5MRETELSk0eaLiXPguBX7/wjGu3w2ufw+Cahhbl4iIiBtTaPI0Gb87Dscd3QkmL7j6Obj8MTCbja5MRETErSk0eQq7HX6dAgtGgLUYQuLgpilQu4PRlYmIiHgEhSZPUJTjuLP35lmO8SW9HA/bDYwwti4REREPotDk7tI3wJeD4PhuMHtD9zHQMRlMJqMrExER8SgKTe7Kboc178Oi58BaAqG1HYfj4tsZXZmIiIhHUmhyR4XH4Ztk2PadY9z4GrjuHQgIN7YuERERD+Z2l1QdO3aMO++8k5CQEMLCwrj33nvJy8s74zZdunTBZDKVeQ0dOrTMOvv27aNv374EBgYSFRXFU089RWlpaUW2cn4OrIP3rnQEJrMP9HoFbv1UgUlERKSCud2epjvvvJODBw+SmpqKxWJh0KBB3H///UyfPv2M2w0ZMoRx48Y5x4GBgc7fW61W+vbtS0xMDCtWrODgwYPcc889+Pj48NJLL1VYL+fEbse8+v/gx3FgK4XwBLhpKsRdZnRlIiIiFwW3Ck1bt25lwYIFrF27lrZt2wLw9ttv06dPH15//XViY2NPu21gYCAxMTGnfG/RokVs2bKFH374gejoaFq1asXzzz/P008/zZgxY/D19a2Qfsqt8Djt/5yI18YNjnGT6+Dat8E/1Ni6RERELiJuFZpWrlxJWFiYMzABdO/eHbPZzOrVq7n++utPu+1nn33Gp59+SkxMDP369WPkyJHOvU0rV66kefPmREf/7wG2SUlJPPjgg2zevJnWrVufcs7i4mKKi4ud45ycHAAsFgsWi+WCenU6vhuv/1xHzdx07F5+2Ho8j+2yQY6r41z1GQY78bVy2desCvL0Hj29P/D8HtWf+/P0Hiuyv/LO6VahKSMjg6ioqDLLvL29iYiIICMj47Tb3XHHHdSpU4fY2Fg2bdrE008/zfbt25k1a5Zz3r8HJsA5PtO848ePZ+zYsSctX7RoUZnDfxfCZC+lszUAX79o1iYkk5MZA99/75K5q5rU1FSjS6hwnt6jp/cHnt+j+nN/nt5jRfRXUFBQrvWqRGh65plneOWVV864ztatW897/vvvv9/5++bNm1OzZk26devGrl27qF+//nnPO2LECFJSUpzjnJwc4uPj6dmzJyEhIec97z+VdmzNj7+soWuva/Hx8XHZvFWFxWIhNTWVHj16eGR/4Pk9enp/4Pk9qj/35+k9VmR/J44UnU2VCE1PPPEEAwcOPOM69erVIyYmhkOHDpVZXlpayrFjx057vtKpJCYmArBz507q169PTEwMa9asKbNOZmYmwBnn9fPzw8/P76TlPj4+rv2GRsRT6vW76+etYjy9P/D8Hj29P/D8HtWf+/P0Hiuiv/LOVyVCU40aNahRo8ZZ1+vYsSNZWVmsW7eONm3aAPDjjz9is9mcQag8Nm7cCEDNmjWd87744oscOnTIefgvNTWVkJAQmjRpco7diIiIiCdyq/s0XXrppfTq1YshQ4awZs0afvnlF5KTk7ntttucV86lpaXRuHFj556jXbt28fzzz7Nu3Tr27NnDt99+yz333MOVV15JixYtAOjZsydNmjTh7rvv5rfffmPhwoU899xzDBs27JR7kkREROTi41ahCRxXwTVu3Jhu3brRp08fOnfuzPvvv+9832KxsH37dudJXb6+vvzwww/07NmTxo0b88QTT3DjjTcyd+5c5zZeXl589913eHl50bFjR+666y7uueeeMvd1EhERkYtblTg8dy4iIiLOeCPLhIQE7Ha7cxwfH8/SpUvPOm+dOnWYP3++S2oUERERz+N2e5pEREREjKDQJCIiIlIOCk0iIiIi5aDQJCIiIlIOCk0iIiIi5aDQJCIiIlIOCk0iIiIi5aDQJCIiIlIOCk0iIiIi5eB2dwSvyk7ciTwnJ8el81osFgoKCsjJyfHIJ1d7en/g+T16en/g+T2qP/fn6T1WZH8n/t3++xNFTkWhyYVyc3MBx6NbRERExL3k5uYSGhp62vdN9rPFKik3m81Geno6wcHBmEwml82bk5NDfHw8+/fvJyQkxGXzVhWe3h94fo+e3h94fo/qz/15eo8V2Z/dbic3N5fY2FjM5tOfuaQ9TS5kNpupVatWhc0fEhLikX8QTvD0/sDze/T0/sDze1R/7s/Te6yo/s60h+kEnQguIiIiUg4KTSIiIiLloNDkBvz8/Bg9ejR+fn5Gl1IhPL0/8PwePb0/8Pwe1Z/78/Qeq0J/OhFcREREpBy0p0lERESkHBSaRERERMpBoUlERESkHBSaRERERMpBoclNFRcX06pVK0wmExs3bjS6HJe69tprqV27Nv7+/tSsWZO7776b9PR0o8tyiT179nDvvfdSt25dAgICqF+/PqNHj6akpMTo0lzmxRdfpFOnTgQGBhIWFmZ0OS4xadIkEhIS8Pf3JzExkTVr1hhdksssW7aMfv36ERsbi8lkYs6cOUaX5FLjx4+nXbt2BAcHExUVRf/+/dm+fbvRZbnMu+++S4sWLZw3fOzYsSPff/+90WVVmJdffhmTycRjjz1myOcrNLmp4cOHExsba3QZFaJr16588cUXbN++na+//ppdu3Zx0003GV2WS2zbtg2bzcZ7773H5s2beeONN5g8eTLPPvus0aW5TElJCTfffDMPPvig0aW4xMyZM0lJSWH06NGsX7+eli1bkpSUxKFDh4wuzSXy8/Np2bIlkyZNMrqUCrF06VKGDRvGqlWrSE1NxWKx0LNnT/Lz840uzSVq1arFyy+/zLp16/j111+5+uqrue6669i8ebPRpbnc2rVree+992jRooVxRdjF7cyfP9/euHFj++bNm+2AfcOGDUaXVKG++eYbu8lkspeUlBhdSoV49dVX7XXr1jW6DJebOnWqPTQ01OgyLlj79u3tw4YNc46tVqs9NjbWPn78eAOrqhiAffbs2UaXUaEOHTpkB+xLly41upQKEx4ebv/www+NLsOlcnNz7Q0bNrSnpqbar7rqKvujjz5qSB3a0+RmMjMzGTJkCP/5z38IDAw0upwKd+zYMT777DM6deqEj4+P0eVUiOzsbCIiIowuQ06hpKSEdevW0b17d+cys9lM9+7dWblypYGVyfnKzs4G8Mg/c1arlRkzZpCfn0/Hjh2NLselhg0bRt++fcv8WTSCQpMbsdvtDBw4kKFDh9K2bVujy6lQTz/9NNWqVaN69ers27ePb775xuiSKsTOnTt5++23eeCBB4wuRU7hyJEjWK1WoqOjyyyPjo4mIyPDoKrkfNlsNh577DEuv/xymjVrZnQ5LvP7778TFBSEn58fQ4cOZfbs2TRp0sToslxmxowZrF+/nvHjxxtdikJTVfDMM89gMpnO+Nq2bRtvv/02ubm5jBgxwuiSz1l5ezzhqaeeYsOGDSxatAgvLy/uuece7FX45vXn2h9AWloavXr14uabb2bIkCEGVV4+59OfSFUzbNgw/vvf/zJjxgyjS3GpRo0asXHjRlavXs2DDz7IgAED2LJli9FlucT+/ft59NFH+eyzz/D39ze6HD1GpSo4fPgwR48ePeM69erV45ZbbmHu3LmYTCbncqvVipeXF3feeScff/xxRZd63srbo6+v70nLDxw4QHx8PCtWrKiyu5zPtb/09HS6dOlChw4dmDZtGmZz1f7/y/l8/6ZNm8Zjjz1GVlZWBVdXcUpKSggMDOSrr76if//+zuUDBgwgKyvL4/aAmkwmZs+eXaZXT5GcnMw333zDsmXLqFu3rtHlVKju3btTv3593nvvPaNLuWBz5szh+uuvx8vLy7nMarViMpkwm80UFxeXea+ieVfaJ8lp1ahRgxo1apx1vbfeeosXXnjBOU5PTycpKYmZM2eSmJhYkSVesPL2eCo2mw1w3GahqjqX/tLS0ujatStt2rRh6tSpVT4wwYV9/9yZr68vbdq0YfHixc4gYbPZWLx4McnJycYWJ+Vit9t5+OGHmT17NkuWLPH4wASOn9Gq/PfluejWrRu///57mWWDBg2icePGPP3005UamEChya3Url27zDgoKAiA+vXrU6tWLSNKcrnVq1ezdu1aOnfuTHh4OLt27WLkyJHUr1+/yu5lOhdpaWl06dKFOnXq8Prrr3P48GHnezExMQZW5jr79u3j2LFj7Nu3D6vV6ryPWIMGDZw/s+4kJSWFAQMG0LZtW9q3b8/EiRPJz89n0KBBRpfmEnl5eezcudM53r17Nxs3biQiIuKkv3Pc0bBhw5g+fTrffPMNwcHBznPRQkNDCQgIMLi6CzdixAh69+5N7dq1yc3NZfr06SxZsoSFCxcaXZpLBAcHn3T+2YnzXQ05L82Qa/bEJXbv3u1xtxzYtGmTvWvXrvaIiAi7n5+fPSEhwT506FD7gQMHjC7NJaZOnWoHTvnyFAMGDDhlfz/99JPRpZ23t99+2167dm27r6+vvX379vZVq1YZXZLL/PTTT6f8fg0YMMDo0lzidH/epk6danRpLjF48GB7nTp17L6+vvYaNWrYu3XrZl+0aJHRZVUoI285oHOaRERERMqh6p9MISIiIlIFKDSJiIiIlINCk4iIiEg5KDSJiIiIlINCk4iIiEg5KDSJiIiIlINCk4iIiEg5KDSJiIiIlINCk4iIiEg5KDSJiIiIlINCk4iIiEg5KDSJiJzF6NGjad68OdWqVSM6OpoHH3wQi8VidFkiUsm8jS5ARKQqs9vt2O123nvvPeLi4tiyZQsDBgygRYsWPPjgg0aXJyKVyGS32+1GFyEi4k7uuOMOoqKimDhxotGliEgl0uE5EZEz2Lt3L8OGDaNZs2aEh4cTFBTEF198Qa1atYwuTUQqmUKTiMhpHD58mHbt2nH06FEmTJjA8uXLWbFiBWazmZYtWxpdnohUMp3TJCJyGnPnzsVqtfL5559jMpkAeOedd7BYLLRq1crY4kSk0ik0iYicRvXq1cnJyeHbb7+lSZMmzJ07l/HjxxMXF0eNGjWMLk9EKplOBBcROQ2bzcZDDz3E9OnTCQgI4K677qKoqIi9e/fy3XffGV2eiFQyhSYRERGRctCJ4CIiIiLloNAkIiIiUg4KTSIiIiLloNAkIiIiUg4KTSIiIiLloNAkIiIiUg4KTSIiIiLloNAkIiIiUg4KTSIiIiLloNAkIiIiUg4KTSIiIiLl8P8pIDHX4ganOwAAAABJRU5ErkJggg==",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(a, torch.sigmoid(a), label=r\"$\\sigma(a)$)\")\n",
    "plt.plot(a, 0.5 + 0.25 * a, label=\"Taylor approx.\")\n",
    "plt.grid(True)\n",
    "plt.xlabel(r\"$a$\")\n",
    "plt.ylabel(r\"$\\sigma(a)$\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cab9e463",
   "metadata": {},
   "source": [
    "Then under this model,\n",
    "\\begin{align*}\n",
    "\\E[Y \\mid \\mbX=\\mbx] \n",
    "&\\approx \\frac{1}{2} + \\frac{\\mbbeta^\\top \\mbx}{4} \n",
    "\\end{align*}\n",
    "or equivalently,\n",
    "\\begin{align*}\n",
    "\\E[Z \\mid \\mbX=\\mbx] \n",
    "&\\approx \\mbbeta^\\top \\mbx\n",
    "\\end{align*}\n",
    "where $Z = 4 (Y - \\tfrac{1}{2}) \\in \\{-2, +2\\}$ is an adjusted response variable. \n",
    "\n",
    "Given a set of $n$ observations of features and (adjusted) responses, $\\{\\mbx_i, z_i\\}_{i=1}^n$, the OLS estimate is,\n",
    "\\begin{align*}\n",
    "\\hat{\\mbbeta}_{\\textsf{OLS}} &= \\left( \\sum_{i=1}^n \\mbx_i \\mbx_i^\\top \\right)^{-1} \\left(\\sum_{i=1}^n \\mbx_i z_i \\right)\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b1f5f847",
   "metadata": {},
   "source": [
    "### Demo\n",
    "\n",
    "Let's try it out for a simulated dataset with scalar covariates $X_i \\sim \\mathrm{N}(0,1)$. We'll simulate responses $Y_i \\mid X_i = x_i \\sim \\mathrm{Bern}(\\sigma(\\beta^* x_i))$ for a few values of $\\beta^*$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "222c3722",
   "metadata": {},
   "outputs": [],
   "source": [
    "def sample_and_fit_ols(seed, n, beta_true):\n",
    "    torch.manual_seed(seed)\n",
    "    X = Normal(0, 1.0).sample((n,))                   # covariates\n",
    "    Y = Bernoulli(logits=X * beta_true).sample()    # binary outcomes\n",
    "\n",
    "    # Fit OLS\n",
    "    Z = 4 * (Y - 0.5)\n",
    "    beta_ols = torch.sum(X * Z) / torch.sum(X ** 2)\n",
    "    return X, Y, beta_ols"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "id": "3cd2b20f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x3379b3dc0>"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Generate, fit, and plot a synthetic dataset\n",
    "seed = 305+ord(\"b\")\n",
    "n = 50\n",
    "beta_true = 1.0\n",
    "X, Y, beta_ols = sample_and_fit_ols(seed, n, beta_true)\n",
    "\n",
    "xs = torch.linspace(-4, 4, 100)\n",
    "plt.plot(X, Y, \"ko\", label=r\"$y_i$\")\n",
    "plt.plot(xs, torch.sigmoid(xs * beta_true), label=r\"$\\beta^*$\")\n",
    "plt.plot(xs, torch.sigmoid(xs * beta_ols), label=r\"$\\hat{\\beta}_{\\mathsf{OLS}}$\")\n",
    "plt.xlabel(r\"$x$\")\n",
    "plt.ylabel(r\"$\\mathbb{E}[Y \\mid X=x; \\beta]$\")\n",
    "plt.legend(loc=\"lower right\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c62a2fa9",
   "metadata": {},
   "source": [
    "Not too shabby! Let's try it with a bunch of simulated datasets."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "id": "560b176f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Density')"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "beta_olss = [sample_and_fit_ols(seed, n, beta_true)[-1] for seed in range(1000)]\n",
    "\n",
    "plt.hist(beta_olss, bins=30, density=True, alpha=0.5, edgecolor=\"k\")\n",
    "plt.axvline(beta_true, color=\"r\", linestyle=\"--\", label=r\"$\\beta^*$\")\n",
    "plt.xlabel(r\"$\\hat{\\beta}_{\\mathsf{OLS}}$\")\n",
    "plt.ylabel(\"Density\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3d397d85",
   "metadata": {},
   "source": [
    "It looks a bit biased, but it's not terrible. \n",
    "\n",
    "What if we try for other values of $\\beta^*$?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "id": "3865720d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1200x400 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 50\n",
    "\n",
    "fig, axs = plt.subplots(1, 3, figsize=(12, 4))\n",
    "for ax, beta_true in zip(axs, [0.1, 1.0, 2.0]):\n",
    "    beta_olss = [sample_and_fit_ols(seed, n, beta_true)[-1] for seed in range(1000)]\n",
    "\n",
    "    ax.hist(beta_olss, bins=20, density=True, alpha=0.5, edgecolor=\"k\")\n",
    "    ax.axvline(beta_true, color=\"r\", linestyle=\"--\", label=r\"$\\beta^*$\")\n",
    "    ax.set_xlabel(r\"$\\hat{\\beta}_{\\mathsf{OLS}}$\")\n",
    "    ax.set_xlim(-3, 3)\n",
    "    ax.set_ylabel(\"Density\")\n",
    "    ax.set_title(rf\"$\\beta^* = {beta_true}$\")\n",
    "plt.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5f5d4171",
   "metadata": {},
   "source": [
    "Uh oh... 😬 \n",
    "\n",
    ":::{admonition} Question\n",
    ":class: tip\n",
    "\n",
    "Why do you think the OLS estimator becomes more biased as $\\beta^*$ grows?\n",
    "::: "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "07ddd89c",
   "metadata": {},
   "source": [
    "## Maximum Likelihood Estimation\n",
    "\n",
    "Let's try a more principled approach: maximum likelihood estimation. For standard linear regression with independent Gaussian responses and homoskedastic noise, $\\hat{\\mbbeta}_{\\mathsf{MLE}} = \\hat{\\mbbeta}_{\\mathsf{OLS}}$. Unfortunately, for logistic regression, there isn't a closed form for $\\hat{\\mbbeta}_{\\mathsf{MLE}}$. However, we can use standard optimization techniques to compute it.\n",
    "\n",
    "There are plenty of standard implementations of maximum likelihood estimation for logistic regression models. Let's see how scikit-learn's implementation fares on the simulated datasets above."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1dc517af",
   "metadata": {},
   "source": [
    "### Demo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "id": "3dc7e562",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sklearn.linear_model import LogisticRegression\n",
    "\n",
    "def sample_and_fit_mle(seed, n, beta_true):\n",
    "    torch.manual_seed(seed)\n",
    "    X = Normal(0, 1.0).sample((n,))                   # covariates\n",
    "    Y = Bernoulli(logits=X * beta_true).sample()      # binary outcomes\n",
    "\n",
    "    # Fit MLE\n",
    "    beta_mle = LogisticRegression().fit(X.reshape(-1, 1), Y).coef_[0, 0]\n",
    "    return X, Y, beta_mle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "id": "c2a98478",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 1200x400 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 50 \n",
    "\n",
    "fig, axs = plt.subplots(1, 3, figsize=(12, 4))\n",
    "for ax, beta_true in zip(axs, [0.1, 1.0, 2.0]):\n",
    "    beta_mles = [sample_and_fit_mle(seed, n, beta_true)[-1] for seed in range(1000)]\n",
    "\n",
    "    ax.hist(beta_mles, bins=20, density=True, alpha=0.5, edgecolor=\"k\")\n",
    "    ax.axvline(beta_true, color=\"r\", linestyle=\"--\", label=r\"$\\beta^*$\")\n",
    "    ax.set_xlabel(r\"$\\hat{\\beta}_{\\mathsf{OLS}}$\")\n",
    "    ax.set_xlim(-3, 3)\n",
    "    ax.set_ylabel(\"Density\")\n",
    "    ax.set_title(rf\"$\\beta^* = {beta_true}$\")\n",
    "plt.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9fde322c",
   "metadata": {},
   "source": [
    "Much better!\n",
    "\n",
    ":::{admonition} Exercise\n",
    ":class: tip\n",
    "\n",
    "Try fitting the MLE for different numbers of data points, $n$. Is it asymptotically biased?\n",
    "::: "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3f25832e",
   "metadata": {},
   "source": [
    "### Deriving the MLE\n",
    "\n",
    "Maximizing the likelihood is equivalent to minimizing the (average) negative log likelihood for a collection of covariates and responses, $\\{\\mbx_i, y_i\\}_{i=1}^n$,\n",
    "\\begin{align*}\n",
    "\\cL(\\mbbeta) \n",
    "&= - \\frac{1}{n} \\sum_{i=1}^n \\log \\mathrm{Bern}(y_i; \\pi(\\mbx_i)) \\\\\n",
    "&= - \\frac{1}{n} \\sum_{i=1}^n y_i \\log \\pi(\\mbx_i) + (1 - y_i) \\log (1 - \\pi(\\mbx_i)) \\\\\n",
    "&= - \\frac{1}{n} \\sum_{i=1}^n y_i \\log \\frac{\\pi(\\mbx_i)}{1 - \\pi(\\mbx_i)} + \\log (1 - \\pi(\\mbx_i)).\n",
    "\\end{align*}\n",
    "Now let's plug in the definition of $\\pi(\\mbx)$. The first term is just the log odds, which we already showed is equal to $\\mbbeta^\\top \\mbx$, the linear component of the model. The second term simplifies too.\n",
    "\\begin{align*}\n",
    "\\cL(\\mbbeta) \n",
    "&= - \\frac{1}{n} \\sum_{i=1}^n \\left[y_i \\mbbeta^\\top \\mbx_i + \\log \\left(1 - \\frac{e^{\\mbbeta^\\top \\mbx_i}}{1 + e^{\\mbbeta^\\top \\mbx_i}} \\right) \\right] \\\\\n",
    "&= - \\frac{1}{n} \\sum_{i=1}^n \\left[y_i \\mbbeta^\\top \\mbx_i - \\log \\left(1 + e^{\\mbbeta^\\top \\mbx_i} \\right) \\right].\n",
    "\\end{align*}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ed800462",
   "metadata": {},
   "source": [
    "\n",
    "### Computing the Gradient\n",
    "We want to maximize the log likelihood, which is of course equivalent to minimizing the negative log likelihood, so let's take the gradient,\n",
    "\\begin{align*}\n",
    "\\nabla \\cL(\\mbbeta) \n",
    "&= -\\frac{1}{n} \\sum_{i=1}^n \\left(y_i \\mbx_i - \\frac{e^{\\mbbeta^\\top \\mbx_i}}{1 + e^{\\mbbeta^\\top \\mbx_i}} \\mbx_i \\right) \\\\\n",
    "&= -\\frac{1}{n}  \\sum_{i=1}^n \\left(y_i - \\sigma(\\mbbeta^\\top \\mbx_i) \\right) \\mbx_i.\n",
    "\\end{align*}\n",
    "Thus, the gradient is a weighted sum of the covariates, and the weights are the residuals $y_i - \\sigma(\\mbx_i^\\top \\mbbeta)$, i.e., the difference between the observed and expected response. \n",
    "\n",
    "This is pretty intuitive! Remember that the gradient points in the direction of steepest descent. This tells us that to increase the log likelihood the most (equivalently, decrease $\\cL$ the most), we should move the coefficient in the direction of covariates where the residual is positive (we are underestimating the mean), and we should move opposite the direction of covariates where the residual is negative (where we are overestimating the mean). \n",
    "\n",
    "Now that we have a closed-form expression for the gradient, we can implement a simple gradient descent algorithm to minimize the negative log likelihood,\n",
    "\\begin{align*}\n",
    "\\mbbeta_{t+1} &\\leftarrow \\mbbeta_t - \\alpha_t \\nabla \\cL(\\mbbeta_t),\n",
    "\\end{align*}\n",
    "where $\\alpha_t \\in \\reals_+$ is the step-size at iteration $i$ of the algorithm. If the step sizes are chosen appropriately and the objective is well behaved, the alorithm converges to at least a local optimum of the log likelihood. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd659c3b",
   "metadata": {},
   "source": [
    "\n",
    "### Convexity of the Log Likelhood\n",
    "\n",
    "If the objective is convex, then all local optima are also global optima, and we can give stronger guarantees on gradient descent. To check the convexity of the log likelihood, we need to compute its Hessian,\n",
    "\\begin{align*}\n",
    "\\nabla^2 \\cL(\\mbbeta) \n",
    "&= \\frac{1}{n} \\sum_{i=1}^n \\sigma'(\\mbbeta^\\top \\mbx_i) \\mbx_i \\mbx_i^\\top\n",
    "\\end{align*}\n",
    "where $\\sigma'(a)$ is the derivative of the logistic function. That is,\n",
    "\\begin{align*}\n",
    "\\sigma'(a) \n",
    "&= \\frac{\\dif}{\\dif a}  \\sigma(a)\\\\\n",
    "&= \\frac{e^a}{(1+e^a)^2} \\\\\n",
    "&= \\sigma(a) (1 - \\sigma(a)).\n",
    "\\end{align*}\n",
    "Plugging this in,\n",
    "\\begin{align*}\n",
    "\\nabla^2 \\cL(\\mbbeta) \n",
    "&= \\frac{1}{n} \\sum_{i=1}^n \\sigma(\\mbbeta^\\top \\mbx_i)(1 - \\sigma(\\mbbeta^\\top \\mbx_i)) \\mbx_i \\mbx_i^\\top \\\\\n",
    "&= \\frac{1}{n} \\sum_{i=1}^n w_i \\mbx_i \\mbx_i^\\top,\n",
    "\\end{align*}\n",
    "where the weights are, $w_i = \\sigma(\\mbbeta^\\top \\mbx_i)(1 - \\sigma(\\mbbeta^\\top \\mbx_i)) = \\Var[Y \\mid \\mbX = \\mbx_i]$. \n",
    "In other words, the Hessian is a weighted sum of outer products of covariates where the weights are equal to the conditional variance. \n",
    "Since variances are non-negative, so are the weights, which implies that the Hessian is **positive semi-definite**, which implies that the negative log likelihood is convex.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e46c59a8",
   "metadata": {},
   "source": [
    "### Example\n",
    "\n",
    "As an example, let's plot the negative log likelihood for a scalar covariate example, as above.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "id": "b7c29fa9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "seed = 305+ord(\"b\")\n",
    "n = 50\n",
    "beta_true = 1.0\n",
    "X, Y, beta_mle = sample_and_fit_mle(seed, n, beta_true)\n",
    "\n",
    "betas = torch.linspace(-4, 5, 100)\n",
    "nlls = -Bernoulli(logits=X[:, None] * betas[None, :]).log_prob(Y[:, None]).mean(axis=0)\n",
    "plt.plot(betas, nlls)\n",
    "plt.axvline(beta_true, color=\"k\", linestyle=\"--\", label=r\"$\\beta^*$\")\n",
    "plt.axvline(beta_mle, color=\"r\", linestyle=\"--\", label=r\"$\\hat{\\beta}_{\\mathsf{MLE}}$\")\n",
    "plt.xlabel(r\"$\\beta$\")\n",
    "plt.xlim(betas[0], betas[-1])\n",
    "plt.ylabel(r\"$\\mathcal{L}(\\beta)$\")\n",
    "plt.legend(loc=\"upper right\")\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "065aea5f",
   "metadata": {},
   "source": [
    "## Regularization\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "99169b1b",
   "metadata": {},
   "source": [
    "\n",
    "### Pathologies in the Separable Regime\n",
    "\n",
    "Suppose the two classes are _linearly separable_, \n",
    "\\begin{align*}\n",
    "\\exists \\mbu \\in \\bbS_{p-1} \\text{ s.t. } \n",
    "\\begin{cases}\n",
    "\\mbu^\\top \\mbx_i > 0 & \\text{ if } y_i = 1 \\\\\n",
    "\\mbu^\\top \\mbx_i < 0 &\\text{ if } y_i = 0\n",
    "\\end{cases}\n",
    "\\end{align*}\n",
    "Now let $\\mbbeta = c \\mbu$ for any $c \\in \\reals_+$. We have \n",
    "\\begin{align*}\n",
    "\\lim_{c \\to \\infty} \\sigma(c \\mbu^\\top \\mbx_i) &= y_i.\n",
    "\\end{align*}\n",
    "In this limit, the model is saturated: $\\Pr(y_i \\mid \\mbx_i) = 1$ for all $i=1,\\ldots,n$; the negative log likelihood goes to $\\lim_{c \\to \\infty} \\cL(c \\mbu) = 0$; and the MLE does not exist since $\\mbbeta^\\star$ diverges. \n",
    "\n",
    "If we run gradient descent in this setting, the magnitude of the estimate $\\hat{\\mbbeta}$ will grow without bound, but the objective should converge to zero. Nevertheless, the diverging MLE is problematic for our convergence rate analysis, since it depends on the norm $\\|\\mbbeta^\\star - \\mbbeta_0\\|_2^2$. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9d9b964d",
   "metadata": {},
   "source": [
    "### $L_2$-Regularization\n",
    "\n",
    "These pathologies can be averted with a little regularization,\n",
    "\\begin{align*}\n",
    "\\cL(\\mbbeta) \n",
    "&= - \\frac{1}{n} \\sum_{i=1}^n \\log \\mathrm{Bern}(y_i; \\sigma(\\mbbeta^\\top \\mbx_i)) + \\textcolor{red}{\\frac{\\gamma}{2} \\|\\mbbeta\\|_2^2}.\n",
    "\\end{align*}\n",
    "The regularizer penalizes larger values of the weights, $\\mbbeta$, and the **hyperparameter** $\\gamma \\in \\reals_+$ sets the strength of the penalty. Even in the linearly separable regime, the maximizer is finite.\n",
    "\n",
    "Now, the gradient and Hessian are,\n",
    "\\begin{align*}\n",
    "\\nabla \\cL(\\mbbeta) \n",
    "&= -\\frac{1}{n}  \\sum_{i=1}^n \\left(y_i - \\sigma(\\mbx_i^\\top \\mbbeta) \\right) \\mbx_i \\textcolor{red}{+ \\gamma \\mbbeta} \\\\\n",
    "\\nabla^2 \\cL(\\mbbeta) \n",
    "&= \\frac{1}{n} \\sum_{i=1}^n w_i \\mbx_i \\mbx_i^\\top \\textcolor{red}{+ \\gamma \\mbI}. \n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7797c209",
   "metadata": {},
   "source": [
    "\n",
    "### Choosing the Hyperparameter\n",
    "It remains to select a value of $\\gamma$. There are many ways to do so. One approach is to _not_ choose a single value and instead try to compute the **regularization path**; i.e., the solution $\\hat{\\mbbeta}(\\gamma)$ for a range of $\\gamma \\in [0, \\infty)$.  Another is to hold out a fraction of data and use cross-validation to select the hyperparameter setting that yields the best performance on the held-out data.  \n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f2501c60",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "### Bayesian Perspective\n",
    "From a Bayesian perspective, we can think of the regularizer as a **prior log probability**. In the case above, the regularizer corresponds to a spherical Gaussian prior,\n",
    "\\begin{align*}\n",
    "\\mbbeta &\\sim \\mathrm{N}(\\mbzero, (\\gamma n)^{-1} \\mbI),\n",
    "\\end{align*}\n",
    "where **precision (inverse covariance)** $\\gamma n \\mbI$.  Minimizing the objective above corresponds to doing **maximum a posteriori (MAP)** estimation in the Bayesian model. If you are going to take a Bayesian stance, it's a bit strange to summarize the entire posterior by its mode, though. We are interested in the posterior distribution as a whole. At the very least, we'd like to know the variance around the mode. We'll talk more about Bayesian methods in the coming weeks.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5ae6388e",
   "metadata": {},
   "source": [
    "## Newton's Method\n",
    "Gradient descent leverages the gradient at $\\mbbeta$ to determine the update. Newton's method uses the Hessian to inform the update as well, and in doing so it can achieve considerably faster convergence.\n",
    "\n",
    "The second order Taylor approximation of $\\cL$ around $\\mbbeta$ is \n",
    "\\begin{align*}\n",
    "\\cL(\\mbbeta') \\approx \\cL(\\mbbeta) + \\nabla \\cL(\\mbbeta)^\\top (\\mbbeta' - \\mbbeta) + \\frac{1}{2} (\\mbbeta' - \\mbbeta)^\\top \\nabla^2 \\cL(\\mbbeta) (\\mbbeta' - \\mbbeta) \\triangleq \\widetilde{\\cL}(\\mbbeta')\n",
    "\\end{align*}\n",
    "The stationary point of $\\widetilde{\\cL}(\\mbbeta')$ is at \n",
    "\\begin{align*}\n",
    "\\nabla \\widetilde{\\cL}(\\mbbeta') &= \\nabla \\cL(\\mbbeta) + \\nabla^2 \\cL(\\mbbeta) (\\mbbeta' - \\mbbeta) = 0 \\\\\n",
    "\\implies \\mbbeta' &= \\mbbeta - [\\nabla^2 \\cL(\\mbbeta)]^{-1} \\nabla \\cL(\\mbbeta),\n",
    "\\end{align*}\n",
    "assuming the Hessian is invertible. When $\\nabla^2 \\cL(\\mbbeta) \\succ 0$ &mdash; i.e., when the Hessian is positive definite &mdash; the inverse Hessian exists and the stationary point is a minimizer. \n",
    "\n",
    "The vanilla Newton's method applies this update repeatedly until convergence, forming a quadratic approximation and then minimizing it. _Damped_ Newton's method adds a step size $\\alpha_t < 1$ to improve stability, \n",
    "\\begin{align*}\n",
    "\\mbbeta_{t+1} &\\leftarrow \\mbbeta_t - \\alpha_t [\\nabla^2 \\cL(\\mbbeta_t)]^{-1} \\nabla \\cL(\\mbbeta_t),\n",
    "\\end{align*}\n",
    "and the step size can be chosen by backtracking line search, for example. \n",
    "\n",
    ":::{admonition} Question\n",
    "Compare the Newton update to that of gradient descent. How do they differ? \n",
    ":::\n"
   ]
  },
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   "cell_type": "markdown",
   "id": "a269b214",
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    "\n",
    "## Iteratively Reweighted Least Squares\n",
    "\n",
    "Newton's method solves for the minimum of a quadratic approximation to the loss function at each iteration. What else involves minimizing a quadratic loss function? Least squares. It turns out that Newton's method can be viewed as iteratively solving a weighted least squares problem. To see this, Let's first write the gradient and Hessian in matrix form,\n",
    "\\begin{align*}\n",
    "\\nabla \\cL(\\mbbeta_t) &= -\\frac{1}{n} \\mbX^\\top (\\mby - \\hat{\\mby}_t) \\\\\n",
    "\\nabla^2 \\cL(\\mbbeta_t) &= \\frac{1}{n} \\mbX^\\top \\mbW_t \\mbX\n",
    "\\end{align*}\n",
    "where \n",
    "- $\\mbX \\in \\reals^{n \\times p}$ is the design matrix with rows $\\mbx_i^\\top$\n",
    "- $\\mby \\in \\{0,1\\}^n$ is the vector of binary responses\n",
    "- $\\hat{\\mby}_t = \\sigma(\\mbX \\mbbeta_t) \\in [0,1]^n$ is the vector of predicted response means\n",
    "- $\\mbW_t = \\diag([w_{t,1}, \\ldots, w_{t,n}])$ with $w_{t,i} = \\sigma(\\mbx_i^\\top \\mbbeta_t) (1- \\sigma(\\mbx_i^\\top \\mbbeta_t))$ is the diagonal weight matrix, where the weights are given by the conditional variances.\n",
    "\n",
    "Then the (undamped) Newton update is,\n",
    "\\begin{align*}\n",
    "\\mbbeta_{t+1} \n",
    "&= \\mbbeta_t - [\\nabla^2 \\cL(\\mbbeta_t)]^{-1} \\nabla \\cL(\\mbbeta_t) \\\\\n",
    "&= \\mbbeta_t + [\\mbX^\\top \\mbW_t \\mbX]^{-1} \\mbX^\\top (\\mby - \\hat{\\mby}_t) \\\\\n",
    "&= [\\mbX^\\top \\mbW_t \\mbX]^{-1} \\mbX^\\top \\mbW_t \\mbX \\mbbeta_t + [\\mbX^\\top \\mbW_t \\mbX]^{-1} \\mbX^\\top (\\mby - \\hat{\\mby}_t) \\\\\n",
    "&= [\\mbX^\\top \\mbW_t \\mbX]^{-1} \\mbX^\\top \\mbW_t \\mbz_t\n",
    "\\end{align*}\n",
    "where \n",
    "\\begin{align*}\n",
    "\\mbz_t &= \\mbX \\mbbeta_t + \\mbW_t^{-1} (\\mby - \\hat{\\mby}_t).\n",
    "\\end{align*}\n",
    "We recognize the update the update above as the solution to a **weighted least squares** problem with **weights** $w_{t,i}$ and **adjusted (or working) responses** $z_{t,i}$.\n",
    "\n",
    "How can we interpret the working responses? We can view them as the real responses mapped through a Taylor approximation of the link (inverse mean) function,\n",
    "\\begin{align*}\n",
    "\\sigma^{-1}(y_i) \n",
    "&\\approx \\sigma^{-1}(\\hat{y}_{t,i}) + (y_i - \\hat{y}_{t,i}) [\\sigma^{-1}]'(\\hat{y}_{t,i}) \\\\\n",
    "&= \\mbx_i^\\top \\mbbeta_t + \\frac{(y_i - \\hat{y}_{t,i})}{w_{t,i}} \\\\\n",
    "&\\triangleq z_{t,i}.\n",
    "\\end{align*}\n",
    "\n"
   ]
  },
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   "cell_type": "markdown",
   "id": "7f0aa878",
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   "source": [
    "\n",
    "## Asymptotic Covariance of MLE\n",
    "\n",
    "What other insight can we glean from the Hessian? Recall our discussion of the asymptotic normality of the MLE from [Lecture 1](01_distributions.ipynb). For iid observations $Y_i \\iid{\\sim} p(\\cdot; \\mbtheta)$ for $i=1,\\ldots,n$, the asymptotic covariance is $\\cI(\\mbtheta)^{-1} / n$, where\n",
    "\\begin{align*}\n",
    "\\cI(\\mbtheta)= -\\E[\\nabla_{\\mbtheta}^2 \\log p(Y; \\mbtheta)]\n",
    "\\end{align*} \n",
    "is the Fisher information matrix. \n",
    "\n",
    "For logistic regression, we have $n$ independent but _not_ identically distributed observations. In this case, the asymptotic covariance follows the same form. It is given by the inverse of the Fisher information matrix; i.e., the inverse of the negative expected Hessian of the log likelihood,\n",
    "\\begin{align*}\n",
    "\\cI(\\mbbeta) = -\\sum_{i=1}^N \\E[\\nabla_{\\mbbeta}^2 \\log p(Y_i \\mid \\mbX_i=\\mbx_i; \\mbbeta)].\n",
    "\\end{align*}\n",
    "(Note that this includes the iid formula as a special case.)\n",
    "\n",
    "Substituting the form of the Hessian from above,\n",
    "\\begin{align*}\n",
    "\\cI(\\mbbeta)\n",
    "&= \\sum_{i=1}^n \\E[\\Var[Y_i \\mid \\mbX = \\mbx_i] \\mbx_i \\mbx_i^\\top] \\\\\n",
    "&= \\sum_{i=1}^n w_i \\mbx_i \\mbx_i^\\top \n",
    "\\end{align*}\n",
    "where $w_i = \\sigma(\\mbbeta^\\top \\mbx_i) (1 - \\sigma(\\mbbeta^\\top \\mbx_i))$. Finally, we evaluate the Fisher information the MLE to obtain the asymptotic covariance estimate,\n",
    "\\begin{align*}\n",
    "\\widehat{\\Cov}(\\hat{\\mbbeta}) &= [\\cI(\\hat{\\mbbeta})]^{-1}.\n",
    "\\end{align*}\n",
    "\n",
    "Like before, we can use the asymptotic covariance estimate to derive Wald confidence intervals for the parameters and perform hypothesis tests. \n",
    "\n",
    "\n",
    ":::{admonition} Caution\n",
    ":class: warning\n",
    "Remember, Wald confidence intervals are only as good as the asymptotic normality assumption. When the likelihod is not well approximated by a quadratic, the covariance estimate will be poor, and the confidence intervals will be invalid. When might the Gaussian approximation not hold?\n",
    ":::\n",
    "\n",
    "\n",
    "<!-- ## Bayesian Inference with the Laplace Approximation\n",
    "\n",
    "Finally, suppose you wanted to perform Bayesian inference of the weights under a Gaussian prior,\n",
    "\\begin{align*}\n",
    "\\mbbeta &\\sim \\mathrm{N}(\\mbzero, \\gamma^{-1} \\mbI).\n",
    "\\end{align*}\n",
    "Unfortunately, the posterior does not have a closed formation solution. Instead, a common form of approximate posterior inference is the **Laplace approximation**,\n",
    "\\begin{align*}\n",
    "p(\\mbbeta \\mid \\mbX, \\mby) &\\approx \\mathrm{N}(\\mbbeta_{\\mathsf{MAP}}, \\widehat{\\mbSigma})\n",
    "\\end{align*}\n",
    "where\n",
    "\\begin{align*}\n",
    "\\cL(\\mbbeta) \n",
    "&= \\log p(\\mbbeta) + \\sum_{i=1}^n \\log p(y_i \\mid \\mbx_i, \\mbbeta) \\\\\n",
    "&= \\log \\mathrm{N}(\\mbbeta; \\mbzero, \\gamma^{-1} \\mbI) + \\sum_{i=1}^n \\log \\mathrm{Bern}(y_i \\mid \\sigma(\\mbx_i^\\top \\mbbeta))\n",
    "\\end{align*}\n",
    "is the log joint probability (_no longer the same as the loss function above!_),\n",
    "\\begin{align*}\n",
    "\\mbbeta_{\\mathsf{MAP}} \n",
    "&= \\arg \\max_{\\mbbeta} \\cL(\\mbbeta)\n",
    "\\end{align*}\n",
    "is the _maximum a posteriori (MAP)_ estimate, and,\n",
    "\\begin{align}\n",
    "\\widehat{\\mbSigma}\n",
    "&= -[\\nabla^2 \\cL(\\mbbeta_{\\mathsf{MAP}})]^{-1} = \\cI(\\mbbeta_{\\mathsf{MAP}})^{-1}\n",
    "\\end{align}\n",
    "is an approximation of the posterior covariance.\n",
    "\n",
    ":::{admonition} Question\n",
    "How do posterior credible intervals under the Laplace approximation compare to Wald confidence intervals of the MLE under L2 regularization? \n",
    ":::\n",
    "\n",
    "### Approximating the model evidence\n",
    "\n",
    "We can use the Laplace approximate to estimate the log marginal likelihood &mdash; a.k.a., the **model evidence**. Note that,\n",
    "\\begin{align*}\n",
    "\\log p(\\mby \\mid \\mbX) \n",
    "&= \\log p(\\mby, \\mbbeta \\mid \\mbX) - \\log p(\\mbbeta \\mid \\mbX, \\mby) \\\\\n",
    "&\\approx \\cL(\\mbbeta) - \\log \\mathrm{N}(\\mbbeta \\mid \\mbbeta_{\\mathsf{MAP}}, \\hat{\\mbSigma}).\n",
    "\\end{align*}\n",
    "The first line is just Bayes' rule, and the right hand side holds for all $\\mbbeta$. The second line is our Laplace approximation.\n",
    "Evaluating at the MAP estimate,\n",
    "\\begin{align*}\n",
    "\\log p(\\mby \\mid \\mbX) \n",
    "&\\approx \\cL(\\mbbeta_{\\mathsf{MAP}}) - \\log \\mathrm{N}(\\mbbeta_{\\mathsf{MAP}} \\mid \\mbbeta_{\\mathsf{MAP}}, \\hat{\\mbSigma}), \\\\\n",
    "&\\approx \\cL(\\mbbeta_{\\mathsf{MAP}}) + \\frac{D}{2} \\log 2 \\pi + \\frac{1}{2} \\log |\\hat{\\mbSigma}| \n",
    "\\end{align*}\n",
    "The marginal likelihood is a natural measure of model complexity, and it is often used for model selection (e.g., determining which features to include in the model). Here, we obtain a simple approximation that incorporates the log probability at the mode along with the \"width\" of the posterior. We'll revisit this approximation when we talk about model selection next week.  -->\n",
    "\n"
   ]
  },
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   "source": [
    "## Convergence Rates\n",
    "\n",
    "Gradient descent is pretty simple to implement. Is it worth our time and effort to implement Newton's method for logistic regression? Often the answer is a resounding yes! As we'll see, Newton's method can achieve second order converge rates, whereas gradient descent is only linear. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bd5f4675",
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   "source": [
    "### Converge Rate of Gradient Descent\n",
    "To determine when and at what rate gradient descent converges, we need to know more about the eigenvalues of the Hessian. \n",
    "\n",
    "If we can bound the maximum eigenvalue of the Hessian by $L$, then we can obtain a quadratic upper bound on the negative log likelihood,\n",
    "\\begin{align*}\n",
    "\\cL(\\mbbeta') &\\leq \\cL(\\mbbeta) + \\nabla \\cL(\\mbbeta)^\\top (\\mbbeta' - \\mbbeta) + \\frac{L}{2} (\\mbbeta' - \\mbbeta)^\\top (\\mbbeta' - \\mbbeta).\n",
    "\\end{align*}\n",
    "That means the negative log likelihood is an $L$-smooth function.\n",
    "\n",
    ":::{admonition} Example: Bounded Covariates\n",
    ":class: tip\n",
    "\n",
    "If the covariates have bounded norm, $\\|\\mbx_i\\|_2 \\leq B$, then we can bound the maximum eigenvalue of the Hessian by,\n",
    "\\begin{align*}\n",
    "\\lambda_{\\mathsf{max}} \n",
    "&= \\max_{\\mbu \\in \\bbS_{p-1}} \\mbu^\\top \\nabla^2 \\cL(\\mbbeta) \\mbu \\\\\n",
    "&= \\max_{\\mbu \\in \\bbS_{p-1}} \\frac{1}{n} \\sum_{i=1}^n w_i \\mbu^\\top \\mbx_i \\mbx_i^\\top \\mbu \\\\\n",
    "&\\leq \\frac{B^2}{4}\n",
    "\\end{align*}\n",
    "since the weights are the conditional variances of Bernoulli random variables, which are at most $\\tfrac{1}{4}$, and since $\\mbu^\\top \\mbx_i \\leq B$ for all unit vectors $\\mbu \\in \\bbS_{p-1}$ (the unit sphere embedded in $\\reals^p$). This isn't meant to be a tight upper bound.\n",
    ":::\n",
    "\n",
    "If we run gradient descent with a constant step size of $\\alpha = 1/L$, then the algorithm converges at a rate of $1/t$, which means that after $t$ iterations\n",
    "\\begin{align*}\n",
    "\\cL(\\mbbeta^\\star) - \\cL(\\mbbeta_t) \\leq \\frac{L}{t} \\|\\mbbeta^\\star - \\mbbeta_0\\|_2^2,\n",
    "\\end{align*}\n",
    "where $\\mbbeta_0$ is the initial setting of the parameters and $\\mbbeta^\\star$ is the global optimum. \n",
    "\n",
    "Put differently, if we want a gap of at most epsilon, we need to run $t \\sim \\cO(1/\\epsilon)$ iterations of gradient descent. Put differently, if we want to reduce $\\epsilon$ by a factor of 100, we need to run around 100 times as many iterations. This is called a **sub-linear convergence** rate.\n"
   ]
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   "id": "ee3c45fa",
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    "\n",
    "### Converge Rate of Gradient Descent with Regularization\n",
    "\n",
    "With regularization, we can also **lower bound** the objective by a quadratic function,\n",
    "\\begin{align*}\n",
    "\\cL(\\mbbeta') &\\geq \\cL(\\mbbeta) + \\nabla \\cL(\\mbbeta)^\\top (\\mbbeta' - \\mbbeta) + \\frac{\\mu}{2} (\\mbbeta' - \\mbbeta)^\\top (\\mbbeta' - \\mbbeta)\n",
    "\\end{align*}\n",
    "for $\\mu > 0$, which means the objective is $\\mu$-**strongly convex**. \n",
    "\n",
    "For twice differentiable objectives, the minimum eigenvalue of Hessian provides such a lower bound, $\\mu = \\lambda_{\\mathsf{min}}$. In particular, we know that minimum eigenvalue of $\\nabla^2 \\cL(\\mbbeta)$ is at least $\\gamma$. (This bound is achieved when the data are linearly separable, the model is saturated, and the conditional variances are all zero.)\n",
    "\n",
    "For a $L$-smooth and $\\mu$-strongly convex function with stepsize $\\alpha = 1/L$, gradient descent has the following convergence guarantee,\n",
    "\\begin{align*}\n",
    "\\cL(\\mbbeta_{t+1}) - \\cL(\\mbbeta^\\star) &\\leq \\left(1 - \\frac{\\mu}{L} \\right) (\\cL(\\mbbeta_t) - \\cL(\\mbbeta^\\star)).\n",
    "\\end{align*}\n",
    "Applying this bound recursively yields that,\n",
    "\\begin{align*}\n",
    "\\cL(\\mbbeta_t) - \\cL(\\mbbeta^\\star) &\\leq \\left(1 - \\frac{\\mu}{L} \\right)^t (\\cL(\\mbbeta_0) - \\cL(\\mbbeta^\\star)).\n",
    "\\end{align*}\n",
    "If we want to find the number of iterations $t$ to bound the gap by at most $\\epsilon$, we need to solve for $t$ in \n",
    "\\begin{align*}\n",
    "\\left(1 - \\frac{\\mu}{L}\\right)^t ( \\mathcal L(\\beta_0) - \\mathcal L(\\beta^*)) \\leq \\epsilon.\n",
    "\\end{align*}\n",
    "This inequality is equivalent to taking the log of both sides\n",
    "\\begin{align*}\n",
    "\t\\log ( \\mathcal L(\\beta_0) - \\mathcal L(\\beta^*)) + t \\log\\left(1 - \\frac{\\mu}{L}\\right) \\leq \\log \\epsilon.\n",
    "\\end{align*}\n",
    "We can further upper bound the LHS by using the inequality $\\log ( 1 - x) \\leq -x$ to get\n",
    "\\begin{align}\n",
    "\t\\log ( \\mathcal L(\\beta_0) - \\mathcal L(\\beta^*))  - \\frac{\\mu t}{L}\\leq \\log \\epsilon.\n",
    "\\end{align}\n",
    "So, we need to run $t \\geq \\frac{L}{\\mu} \\log \\frac{\\cL(\\mbbeta_0) - \\cL(\\mbbeta^\\star)}{\\epsilon} \\sim \\log \\frac{1}{\\epsilon}$ iterations of gradient descent. If we want to reduce $\\epsilon$ by a factor of 100, we only need to run around $\\log 100$ times as many iterations. This is called **linear convergence**.\n"
   ]
  },
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   "cell_type": "markdown",
   "id": "279c7694",
   "metadata": {},
   "source": [
    "### Converge Rate of Newton's Method\n",
    "\n",
    "Under certain conditions &mdash; if the objective is strongly convex, the Hessian is Lipschitz continuous, and we start near enough to the global optimum &mdash; Newton's method achieves **second order convergence**, meaning\n",
    "\\begin{align*}\n",
    "\\|\\mbbeta_{t+1} - \\mbbeta^\\star\\|_2 \\leq \\left(c \\|\\mbbeta_t - \\mbbeta^\\star\\|_2\\right)^2\n",
    "\\end{align*}\n",
    "for some positive constant $c$, provided we start with $\\mbbeta_0$ close enough to $\\mbbeta^\\star$. \n",
    "Applying this bound recursively yields that,\n",
    "\\begin{align*}\n",
    "\\|\\mbbeta_t - \\mbbeta^\\star\\|_2 &\\leq \\left(c \\|\\mbbeta_0 - \\mbbeta^\\star\\|_2 \\right)^{2^t}.\n",
    "\\end{align*}\n",
    "\n",
    "Put differently, if we start with $\\|\\mbbeta_0 - \\mbbeta^\\star\\| < c^{-1}$, then we need $t \\sim \\cO(\\log \\log \\frac{1}{\\epsilon})$ iterations to obtain an error of $\\epsilon$. Since the double log grows incredibly slowly, this statement says that we effectively need a constant number of iterations for Newton's method to converge in this regime.\n",
    "\n",
    "For more information on convergence rates of gradient descent and Newton's method, see {cite:t}`boyd2004convex`, ch. 9.\n",
    "\n",
    "<!-- \n",
    ":::{admonition} Note\n",
    ":class: warning\n",
    "Note that here the convergence is in terms of distance to the optimum, rather than in the value of the objective function. We can recast the convergence of gradient descent in similar terms. \n",
    "::: -->\n"
   ]
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   "source": [
    "\n",
    "## Conclusion\n",
    "\n",
    "One, two, many... with logistic regression, we can begin modeling relationships between binary response variables and (possibly arbitrary-valued) covariates.  Next, we'll see how to expand from logistic regression to generalized linear models for exponential family responses."
   ]
  }
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