Soss: Lightweight Probabilistic Programming in Julia

Chad Scherrer

November 7, 2019

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

?

  • Simple high-level syntax
  • Uses GeneralizedGenerated.jl for flexible staged compilation
  • Model type parameter includes type-level representation of itself
  • Allows specialized code generation for each primitive(model, data)
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}

Theory

Soss

julia> Soss.sourceLogpdf()(m)
quote
    _ℓ = 0.0
    _ℓ += logpdf(HalfCauchy(3), σ)
    _ℓ += logpdf(Normal(0, 5), μ)
    _ℓ += logpdf(Normal(μ, σ) |> iid(N), x)
    return _ℓ
end
@model N begin
    μ ~ Normal(0, 5)
    σ ~ HalfCauchy(3)
    x ~ Normal(μ, σ) |> iid(N)
end
m = @model x begin
    α ~ Normal()
    β ~ Normal()
    σ ~ HalfNormal()
    yhat = α .+ β .* x
    n = length(x)
    y ~ For(n) do j
        Normal(yhat[j], σ)
    end
end
julia> m(x=truth.x)
Joint Distribution
    Bound arguments: [x]
    Variables: [σ, β, α, yhat, n, y]

@model x begin
        σ ~ HalfNormal()
        β ~ Normal()
        α ~ Normal()
        yhat = α .+ β .* x
        n = length(x)
        y ~ For(n) do j
                Normal(yhat[j], σ)
            end
    end

Observed data is not specified yet!

julia> post = dynamicHMC(m(x=truth.x), (y=truth.y,)) |> particles
(σ = 2.02 ± 0.15, β = 2.99 ± 0.19, α = 0.788 ± 0.2)

Posterior distribution

Possible best-fit lines

Start with Data

Sample Parameters|Data

Sample Data|Parameters

Real Data

Replicated Fake Data

Compare

Posterior

Distribution

Predictive

Distribution

pred = predictive(m, :α, :β, :σ)
@model (x, α, β, σ) begin
    yhat = α .+ β .* x
    n = length(x)
    y ~ For(n) do j
        Normal(yhat[j], σ)
    end
end
m = @model x begin
    α ~ Normal()
    β ~ Normal()
    σ ~ HalfNormal()
    yhat = α .+ β .* x
    n = length(x)
    y ~ For(n) do j
        Normal(yhat[j], σ)
    end
end
postpred = map(post) do θ 
    delete(rand(pred(θ)((x=x,))), :n, :x)
end |> particles

predictive makes a new model!

pvals = mean.(truth.y .> postpred.y)

Where we expect the data

Where we see the data

 m2 = @model x begin
     α ~ Normal()
     β ~ Normal()
     σ ~ HalfNormal()
     yhat = α .+ β .* x
     νinv ~ HalfNormal()
     ν = 1/νinv
     n = length(x)
     y ~ For(n) do j
             StudentT(ν,yhat[j],σ)
         end
 end
julia> post2 = dynamicHMC(m2(x=truth.x), (y=truth.y,)) |> particles
( σ = 0.57 ± 0.09, νinv = 0.609 ± 0.14
, β = 2.73 ± 0.073, α = 0.893 ± 0.077)
julia> Soss.sourceRand()(m)
quote
    σ = rand(HalfNormal())
    β = rand(Normal())
    α = rand(Normal())
    yhat = α .+ β .* x
    n = length(x)
    y = rand(For(((j,)->begin
                Normal(yhat[j], σ)
            end), n))
    (x = x, yhat = yhat, n = n
    , α = α, β = β, σ = σ, y = y)
end
@model x begin
    σ ~ HalfNormal()
    β ~ Normal()
    α ~ Normal()
    yhat = α .+ β .* x
    n = length(x)
    y ~ For(n) do j
        Normal(yhat[j], σ)
    end
end
julia> Soss.sourceLogpdf()(m)
quote
    _ℓ = 0.0
    _ℓ += logpdf(HalfNormal(), σ)
    _ℓ += logpdf(Normal(), β)
    _ℓ += logpdf(Normal(), α)
    yhat = α .+ β .* x
    n = length(x)
    _ℓ += logpdf(For(n) do j
                Normal(yhat[j], σ)
            end, y)
    return _ℓ
end
@model x begin
    σ ~ HalfNormal()
    β ~ Normal()
    α ~ Normal()
    yhat = α .+ β .* x
    n = length(x)
    y ~ For(n) do j
        Normal(yhat[j], σ)
    end
end
julia> Soss.sourceSymlogpdf()(m)
quote
    _ℓ = 0.0
    x = sympy.IndexedBase(:x)
    yhat = sympy.IndexedBase(:yhat)
    n = sympy.IndexedBase(:n)
    α = sympy.IndexedBase(:α)
    β = sympy.IndexedBase(:β)
    σ = sympy.IndexedBase(:σ)
    y = sympy.IndexedBase(:y)
    _ℓ += symlogpdf(HalfNormal(), σ)
    _ℓ += symlogpdf(Normal(), β)
    _ℓ += symlogpdf(Normal(), α)
    yhat = sympy.IndexedBase(:yhat)
    n = sympy.IndexedBase(:n)
    _ℓ += symlogpdf(For(n) do j
                Normal(yhat[j], σ)
            end, y)
    return _ℓ
end
@model x begin
    σ ~ HalfNormal()
    β ~ Normal()
    α ~ Normal()
    yhat = α .+ β .* x
    n = length(x)
    y ~ For(n) do j
        Normal(yhat[j], σ)
    end
end
julia> symlogpdf(m)
julia> symlogpdf(m) |> expandSums
-3.7-0.5α^{2}-0.5β^{2}-σ^{2}+\sum_{j_{1}=1}^{n}\left(-0.92-\logσ-\frac{0.5\left(y_{j_{1}}-\hat{y}_{j_{1}}\right)^{2}}{σ^{2}}\right)
-3.7-0.5α^{2}-0.5β^{2}-σ^{2}-0.92n-n\logσ-\frac{0.5}{\sigma^{2}}\sum_{j_{1}=1}^{n}\left(y_{j_{1}}-\hat{y}_{j_{1}}\right)^{2}
julia> symlogpdf(m()) |> expandSums |> foldConstants |> codegen
quote
    var"##add#643" = 0.0
    var"##add#643" += -3.6757541328186907
    var"##add#643" += begin
            var"##mul#644" = 1.0
            var"##mul#644" *= -0.5
            var"##mul#644" *= begin
                    var"##arg1#646" = α
                    var"##arg2#647" = 2
                    var"##symfunc#645" = (Soss._pow)(var"##arg1#646", var"##arg2#647")
                    var"##symfunc#645"
                end
            var"##mul#644"
        end
    var"##add#643" += begin
            var"##mul#648" = 1.0
            var"##mul#648" *= -0.5
            var"##mul#648" *= begin
                    var"##arg1#650" = β
                    var"##arg2#651" = 2
                    var"##symfunc#649" = (Soss._pow)(var"##arg1#650", var"##arg2#651")
                    var"##symfunc#649"
                end
            var"##mul#648"
        end
    var"##add#643" += begin
            var"##mul#652" = 1.0
            var"##mul#652" *= -1.0
            var"##mul#652" *= begin
                    var"##arg1#654" = σ
                    var"##arg2#655" = 2
                    var"##symfunc#653" = (Soss._pow)(var"##arg1#654", var"##arg2#655")
                    var"##symfunc#653"
                end
            var"##mul#652"
        end
    var"##add#643" += begin
            var"##mul#656" = 1.0
            var"##mul#656" *= -0.9189385332046727
            var"##mul#656" *= n
            var"##mul#656"
        end
    var"##add#643" += begin
            var"##mul#657" = 1.0
            var"##mul#657" *= -0.5
            var"##mul#657" *= begin
                    var"##arg1#659" = σ
                    var"##arg2#660" = -2
                    var"##symfunc#658" = (Soss._pow)(var"##arg1#659", var"##arg2#660")
                    var"##symfunc#658"
                end
            var"##mul#657" *= begin
                    let
                        var"##sum#661" = 0.0
                        begin
                            var"##lo#663" = 1
                            var"##hi#664" = n
                            @inbounds for _j1 = var"##lo#663":var"##hi#664"
                                    begin
                                        var"##Δsum#662" = begin
                                                var"##arg1#666" = begin
                                                        var"##add#668" = 0.0
                                                        var"##add#668" += begin
                                                                var"##mul#669" = 1.0
                                                                var"##mul#669" *= -1.0
                                                                var"##mul#669" *= begin
                                                                        var"##arg1#671" = yhat
                                                                        var"##arg2#672" = _j1
                                                                        var"##symfunc#670" = (getindex)(var"##arg1#671", var"##arg2#672")
                                                                        var"##symfunc#670"
                                                                    end
                                                                var"##mul#669"
                                                            end
                                                        var"##add#668" += begin
                                                                var"##arg1#674" = y
                                                                var"##arg2#675" = _j1
                                                                var"##symfunc#673" = (getindex)(var"##arg1#674", var"##arg2#675")
                                                                var"##symfunc#673"
                                                            end
                                                        var"##add#668"
                                                    end
                                                var"##arg2#667" = 2
                                                var"##symfunc#665" = (Soss._pow)(var"##arg1#666", var"##arg2#667")
                                                var"##symfunc#665"
                                            end
                                        var"##sum#661" += var"##Δsum#662"
                                    end
                                end
                        end
                        var"##sum#661"
                    end
                end
            var"##mul#657"
        end
    var"##add#643" += begin
            var"##mul#676" = 1.0
            var"##mul#676" *= -1.0
            var"##mul#676" *= n
            var"##mul#676" *= begin
                    var"##arg1#678" = σ
                    var"##symfunc#677" = (log)(var"##arg1#678")
                    var"##symfunc#677"
                end
            var"##mul#676"
        end
    var"##add#643"
end
julia> @btime logpdf($m(x=x), $truth)
1.911 μs (25 allocations: 1.42 KiB)
-901.7607073245318
julia> @btime logpdf($m(x=x), $truth, codegen)
144.671 ns (1 allocation: 896 bytes)
-903.4977930382969

Default

Code Generation

  • New feature, still in development
  • Speedup depends on lots of things
julia> m = @model begin
           a ~ @model begin
               x ~ Normal()
           end
       end;

julia> rand(m())
(a = (x = -0.20051706307697828,),)
julia> m2 = @model anotherModel begin
           y ~ anotherModel
           z ~ anotherModel
           w ~ Normal(y.a.x / z.a.x, 1)
       end;

julia> rand(m2(anotherModel=m)).w
-1.822683102320004
  • Stream combinators via Transducers.jl and OnlineStats.jl
  • Connection to other PPLs: Turing.jl, Gen.jl
  • Normalizing flows with Bijectors.jl
  • Deep learning with Flux.jl
  • Gaussian processes

Thank You!

https://cscherrer.github.io/

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

julia-vscode

MonteCarloMeasurements.jl

Plots.jl

Distributions.jl

SymPy.jl

DynamicHMC.jl

MLStyle.jl

MacroTools.jl

Special Thanks for

AverageShiftedHistograms.jl

2019-11-07-Soss

By Chad Scherrer

2019-11-07-Soss

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