Path Integral Approach For Functional Priors

PriorOnFuncs.jl

using Pkg
Pkg.activate(".")
Pkg.add(["Turing","Distributions","ApproxFun","QuadGK","StatsPlots","LinearAlgebra","DataFrames"])

using Turing, Distributions, ApproxFun, QuadGK, StatsPlots, LinearAlgebra, DataFrames

# a prior over functions that go up then come down... First let's see how ApproxFun works

Random.set_global_seed!(123456)
f = Fun(Chebyshev(Interval(0..100)),randn(12))

plot(collect(0:100),map(f,0:100))

@model function updown(n)
    c ~ MvNormal(repeat([0.0],n),1000.0^2*I(n))
    ## candidate functions rise from 0 to 1 over 0-20, stay near constant out to 50, 
    ## and decline exponentially ish down to 0.3 out to 100
    ## this could be some kind of function for say immune response as a function
    ## of age in years, or whatever
    f = Fun(Chebyshev(Interval(0..100)),c)
    a = quadgk(x -> ((f(x) - x/20)/.25)^2,0,20) # near linear rise over x 0..20
    Turing.@addlogprob!(-a[1])
    b = quadgk(x -> ((f(x) - 1.0)/.1)^2,20,50) # near 1 for x in 20-50
    Turing.@addlogprob!(-b[1])
    c = quadgk(x -> (((f(x) - (exp(-(x-50)/10)+0.2))/.25)^2),50,100) #exponential decay to 0.2
    Turing.@addlogprob!(-c[1])
    ## relatively smooth functions
    ddf = Derivative(2)*f
    d = quadgk(x -> (ddf(x)^2),0,100)
    Turing.@addlogprob!(-d[1]/100.0/.2)
end

s = Turing.sample(updown(10),NUTS(300,.8),300)

df = DataFrame(s)

p = plot(;legend=false)
x = collect(0.0:100.0)
for i in 1:10:300 
    plot!(x,map(Fun(Chebyshev(Interval(0..100)),Vector(df[i,3:12])),x))
end

display(p)