WIP: Comparing various backwards schemes

[mschauer]

@mmider

[mschauer]

Compute numbers for likelihoods in the model

struct IntegratedDiffusion <: ContinuousTimeProcess{ℝ{2}}
    γ::Float64
end

βu(t, x::Float64, P::IntegratedDiffusion) = - (x+sin(x)) + 1/2
Bridge.b(t::Float64, x, P::IntegratedDiffusion) = ℝ{2}(x[2], βu(t, x[2], P))
Bridge.σ(t, x, P::IntegratedDiffusion) = ℝ{2}(0.0, P.γ)

Bridge.constdiff(::IntegratedDiffusion) = true

struct IntegratedDiffusionAux <: ContinuousTimeProcess{ℝ{2}}
    γ::Float64
end

βu(t, x::Float64, P::IntegratedDiffusionAux) = -x + 1/2
Bridge.b(t::Float64, x, P::IntegratedDiffusionAux) = ℝ{2}(x[2], βu(t, x[2], P))

with parameters

T = 1.5
dt = 1/1000

tt = 0.:dt:T

x0 = ℝ{2}(2.0, 1.0)

L = @SMatrix [1. 0.]
Σ = @SMatrix [0.1]
v = ℝ{1}(2.5)

γ = 0.7

ϵ = 0.00001

which we can compare the FitzHughNagumo code against: e.g

LP = log(pdf(Normal((μt[1])[1] + (Lt[1] * x0)[1], (Mt[1])[1] ^ -0.5), v[1])) 
 -0.9929361410325926

[mschauer]

Gradient descent towards maximum likelihood of the start value:
https://www.dropbox.com/s/2olgz3jg8mqk2nz/gradientdescent.mp4?dl=0

The gradient of the momenta seems to be very low.