mschauer

using CausalInference, Graphs
V = [:U, :T, :P, :O]
ι = Dict(v=>i for (i,v) in enumerate(V))
g = digraph([1=>3, 2=>3, 3=>4, 2=>4, 1=>4])


# Can estimate total effect T=>O without observing U?
u = ι[:T]
v = ι[:O]
∅ = Set{Int}()

mschauer

function restart()
    startup = """
        Base.ACTIVE_PROJECT[]=$(repr(Base.ACTIVE_PROJECT[]))
        Base.HOME_PROJECT[]=$(repr(Base.HOME_PROJECT[]))
        cd($(repr(pwd())))
        """
    cmd = `$(Base.julia_cmd()) -ie $startup`
    atexit(()->run(cmd))
    exit(0)
end

mschauer

using Kalman
using Kalman.GaussianDistributions
using Statistics, LinearAlgebra

# prior for time 0
x0 = 0.0
P0 = floatmax(x0)

# observation operator
H = 1.0

mschauer

using GLMakie
using Random 

# mixture distribution
X(c) = rand() < c ? sqrt(rand()) :  1 - sqrt(rand())

# sample
Random.seed!(1);
A = [0:0.01:1;; X.(0:0.01:1)]; 

mschauer

 0.0 0.0
 -0.032984387220723596 0.030216400626819087
 0.005532055967160396 -0.06303492443179701
 0.04716487617405366 0.06151822897388728
 -0.08816371845816047 -0.01559492041364278
 0.0844812855043274 -0.05374007858781831
 -0.02848102330289795 0.10594798706320867
 -0.054631054889518714 -0.10518877181328493
 0.1190544569384441 0.04347849672678073
 -0.12429436989984163 0.05130690904254036

mschauer

• You must have heard the term detailed balance in the context of Markov chains before.

Yes, I have heard of the term detailed balance in the context of Markov chains. In the context of Markov chains, detailed balance refers to a condition that must be satisfied in order for a given system to be in equilibrium. This condition states that the product of the transition probabilities between any two states in the system must be equal to the product of the reverse transition probabilities between those same two states. This ensures that the system will not tend to move towards one particular state over another, and will instead remain in a state of equilibrium.

• That is not quite correct, let's see. Lets denote the transition probability from state x to state y by p(x → y). Detailed balance holds if there is a probability distribution π(x) over the states such that u(x,y) = π(x)p(x → y) is a symmetric function, hence u(x,y) = u(y,x).

You are correct. My previous explanation was not entirely accurate. Deta

mschauer

struct Dual <: Real
    value
    δ
    # w and Y and tag 
end
Base.show(io::IO, x::Dual) = print(io, x.value, " + ", x.δ, " ϵ")
Base.:+(x::Dual, y::Dual) = Dual(x.value + y.value, x.δ + y.δ)
Base.:*(x::Dual, y::Dual) = Dual(x.value*y.value, x.δ*y.value + x.value*y.δ)
Dual(x) = Dual(x, zero(x))
Base.promote_rule(::Type{Dual},::Type{<:Real}) = Dual

mschauer

d = 100
ts_ = range(0, 1, length=d+1)[2:end]
Γ = SymTridiagonal([2.0ones(d-1); 1.0], -ones(d-1))/ts_[1]

# Note the covariance matrix of B.M is 
ts = range(0, 1, length=d+1)[2:end]
Γ0 = inv([min(i,j) for i in ts_, j in ts_])
@test norm(Γ0 - Γ) < 1e-7

mschauer

using StochasticAD, Distributions, Optimisers, GLMakie
import Random # hide
Random.seed!(1234) # hide

# Function we want to maximize the expectation of
function X(p)
    a = p*(1-p) 
    b = rand(Binomial(10, p))
    c = 2 * b + 3 * rand(Bernoulli(p))
    return a * c * rand(Normal(b, a))

mschauer

# Linear state space system forward filtering backward sampling with Zygote

using LinearAlgebra
using GaussianDistributions
using GaussianDistributions: ⊕, logpdf
using StaticArrays

using Statistics

using Zygote