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

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

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

[deps]
ArraysOfArrays = "65a8f2f4-9b39-5baf-92e2-a9cc46fdf018"
ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
MCMCChains = "c7f686f2-ff18-58e9-bc7b-31028e88f75d"
PDMats = "90014a1f-27ba-587c-ab20-58faa44d9150"
Pathfinder = "b1d3bc72-d0e7-4279-b92f-7fa5d6d2d454"
StructArrays = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
ZigZagBoomerang = "36347407-b186-4a6a-8c98-4f4567861712"

mschauer

# using naive search
using GLMakie, LinearAlgebra
using StaticArrays
rot(θ) = @SMatrix [cos(θ) -sin(θ); sin(θ) cos(θ) ]
x = [Point2f(0,1), Point2f(1,0), Point2f(3, 0)]

xnew = copy(x)
δ = 0.001
xall = [copy(x)]
i = 1

mschauer

#=
ZigZagBoomerang.jl has implemented something like a priority queue keeping track of 
local minima (or high priority task).

Think of graph where vertices are tasks that are assigned priorities (smaller = higher priority) 
and edges between two tasks indicate if the higher priority task has to be worked on before the
lower priority task (edge) or both can be worked on in parallel (no edge).

It’s thread-safe in the sense that priorities can be updated in different threads 
if one has a proper coloring of the vertices and updates the task of one color in @threads