Soss: Lightweight Probabilistic Programming in Julia

Chad Scherrer

Senior Data Scientist, Metis

Served as Technical Lead for language evaluation for the DARPA program
Probabilistic Programming for Advancing Machine Learning (PPAML)

PPL Publications

Managed Uncertainty

Rational Decisions

Bayesian Analysis

Probabilistic Programming

  • Physical systems
  • Hypothesis testing
  • Modeling as simulation
  • Medicine
  • Finance
  • Insurance

Risk

Custom models

https://cscherrer.github.io/post/max-profit/

A disconnect between the "user language" and "developer language"

X

3

Python

C

Deep Learning Framework

  • Harder for beginner users
  • Barrier to entry for developers
  • Limits extensibility

?

  • Models begin with "Tilde code" based on math notation


     
  • A Model is a "lossless" data structure

     

  • Combinators transform code before compilation

Model

Tilde code
\begin{aligned} \mu &\sim \text{Normal}(0,5)\\ \sigma &\sim \text{Cauchy}_+(0,3) \\ x_j &\sim \text{Normal}(\mu,\sigma) \end{aligned}

Advantages

  • Transparent
  • Composable
  • Backend-agnostic
  • No intrinsic overhead
P(\mu,\sigma|x)\propto P(\mu,\sigma)P(x|\mu,\sigma)
\begin{aligned} \mu &\sim \text{Normal}(0,5)\\ \sigma &\sim \text{Cauchy}_+(0,3) \\ x_j &\sim \text{Normal}(\mu,\sigma) \end{aligned}
m = @model N begin
    μ ~ Normal(0, 5)
    σ ~ HalfCauchy(3)
    x ~ Normal(μ, σ) |> iid(N)
end

Theory

Soss

julia> sourceLogdensity(m)
:(function ##logdensity#1168(pars)
      @unpack (μ, σ, x, N) = pars
      ℓ = 0.0
      ℓ += logpdf(Normal(0, 5), μ)
      ℓ += logpdf(HalfCauchy(3), σ)
      ℓ += logpdf(Normal(μ, σ) |> iid(N), x)
      return ℓ
  end)
gaussianMixture = @model begin
    N ~ Poisson(100)
    K ~ Poisson(2.5)
    p ~ Dirichlet(K, 1.0)
    μ ~ Normal(0,1.5) |> iid(K)
    σ ~ HalfNormal(1)
    θ = [(m,σ) for m in μ]
    x ~ MixtureModel(Normal, θ, p) |> iid(N)
end

julia> m = gaussianMixture(N=100,K=2)
@model begin
    p ~ Dirichlet(2, 1.0)
    μ ~ Normal(0, 1.5) |> iid(2)
    σ ~ HalfNormal(1)
    θ = [(m,σ) for m in μ]
    x ~ MixtureModel(Normal, θ, p) |> iid(N)
end
julia> rand(m) |> pairs
pairs(::NamedTuple) with 4 entries:
  :p => 0.9652
  :μ => [-1.20, 3.121]
  :σ => 1.320
  :x => [-1.780, -2.983, -1.629, ...]

rand(m_fwd; groundTruth...)
julia> m_fwd = m(:p,:μ,:σ)
@model (p, μ, σ) begin
    θ = [(m, σ) for m = μ]
    x ~ MixtureModel(Normal,θ,[p,1-p]) |> iid(100)
end
m = @model begin
    p ~ Uniform()
    μ ~ Normal(0, 1.5) |> iid(2)
    σ ~ HalfNormal(1)
    θ = [(m, σ) for m in μ]
    x ~ MixtureModel(Normal,θ,[p,1-p]) |> iid(100)
end
post = nuts(m_inv; groundTruth...)
julia> m_inv = m(:x)
@model x begin
    p ~ Uniform()
    μ ~ Normal(0, 1.5) |> iid(2)
    σ ~ HalfNormal(1)
    θ = [(m, σ) for m = μ]
    x ~ MixtureModel(Normal,θ,[p,1-p]) |> iid(100)
end
m = @model begin
    p ~ Uniform()
    μ ~ Normal(0, 1.5) |> iid(2)
    σ ~ HalfNormal(1)
    θ = [(m, σ) for m in μ]
    x ~ MixtureModel(Normal,θ,[p,1-p]) |> iid(100)
end
x
\theta_1
\theta_2
\theta_n
x^\text{rep}_1
x^\text{rep}_2
x^\text{rep}_2
\vdots
\vdots
x
julia> lda
@model (α, η, K, V, N) begin
    M = length(N)
    β ~ Dirichlet(repeat([η],V)) |> iid(K)
    θ ~ Dirichlet(repeat([α],K)) |> iid(M)
    z ~ For(1:M) do m
            Categorical(θ[m]) |> iid(N[m])
        end
    w ~ For(1:M) do m
            For(1:N[m]) do n
                Categorical(β[(z[m])[n]])
            end
        end
end
julia> dependencies(lda)
               [] => :α
               [] => :η
               [] => :K
               [] => :V
               [] => :N
             [:N] => :M
     [:η, :V, :K] => :β
     [:α, :K, :M] => :θ
     [:M, :θ, :N] => :z
 [:M, :N, :β, :z] => :w

Blei, Ng, & Jordan (2003). Latent Dirichlet Allocation. JMLR, 3(4–5), 993–1022.

m = @model y begin
    μ ~ Normal(0, 1)
    σ ~ HalfCauchy(1)
    ε ~ TDist(5)
    y ~ Normal(μ + ε, σ)
end

symlogpdf(m)
\sum_{j=1}^{N} \left(- \frac{μ^{2}}{2} \\ - \log{\left(σ \right)} \\ - \log{\left(σ^{2} + 1 \right)} \\ - 3 \log{\left(\frac{ε^{2}{\left(j \right)}}{5} + 1 \right)} \\ - 2 \log{\left(\pi \right)} - \frac{\log{\left(5 \right)}}{2} - \log{\left(2 \right)} - \log{\left(\operatorname{B}\left(\frac{1}{2}, \frac{5}{2}\right) \right)} - \frac{\left(- μ + y{\left(j \right)} - ε{\left(j \right)}\right)^{2}}{2 σ^{2}}\right)
-N\left[2\log\left(\pi\right)+\frac{\log\left(5\right)}{2}+\log2+\log\left(B\left(\frac{1}{2},\frac{5}{2}\right)\right)\right]
-N\left[\frac{μ^{2}}{2}+\log\left(σ\right)+\log\left(σ^{2}+1\right)\right]
-\frac{1}{2σ^{2}} \sum_{j=1}^{N}\left(-μ+y\left(j\right)-ε\left(j\right)\right)^{2}
-3\sum_{j=1}^{N}\log\left(\frac{ε\left(j\right)^{2}}{5}+1\right)
p = @model begin
    x ~ Normal()
end

q = @model σ begin
    x ~ HalfNormal(σ)
end

julia> sourceRand(q)
:(function ##rand#726(args...; kwargs...)
      @unpack (σ,) = kwargs
      x = rand(HalfNormal(σ))
      (x = x,)
  end)


julia> sourceImportanceLogWeights(p,q)
:(function ##logimportance#725(pars)
      @unpack (x, σ) = pars
      ℓ = 0.0
      ℓ += logpdf(Normal(), x)
      ℓ -= logpdf(HalfNormal(σ), x)
      return ℓ
  end)
p = @model x begin
    μ ~ Normal()
    x ~ Normal(μ,1) |> iid(20)
end

q = @model m,s begin
    μ ~ Normal(m,s)
end
julia> sourceRand(q)
:(function ##rand#726(args...; kwargs...)
      @unpack (σ,) = kwargs
      x = rand(HalfNormal(σ))
      (x = x,)
  end)


julia> sourceImportanceLogWeights(p,q)
:(function ##logimportance#725(pars)
      @unpack (x, σ) = pars
      ℓ = 0.0
      ℓ += logpdf(Normal(), x)
      ℓ -= logpdf(HalfNormal(σ), x)
      return ℓ
  end)

logo by James Fairbanks, on Julia  Discourse site

JuliaCon 2019

July 21-27

Baltimore, MD

Thank You!

https://cscherrer.github.io/

https://github.com/cscherrer/Soss.jl

2019-05-01 ODSC East

By Chad Scherrer

2019-05-01 ODSC East

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