Running Pathfinder on Turing.jl models
This tutorial demonstrates how Turing can be used with Pathfinder.
We'll demonstrate with a regression example.
using AdvancedHMC, Pathfinder, Random, Turing
Random.seed!(39)
@model function regress(x, y)
α ~ Normal()
β ~ Normal()
σ ~ truncated(Normal(); lower=0)
y .~ Normal.(α .+ β .* x, σ)
end
x = 0:0.1:10
y = @. 2x + 1.5 + randn() * 0.2model = regress(collect(x), y)
n_chains = 88For convenience, pathfinder and multipathfinder can take Turing models as inputs and produce MCMCChains.Chains objects as outputs.
result_single = pathfinder(model; ndraws=1_000)Single-path Pathfinder result
tries: 1
draws: 1000
fit iteration: 22 (total: 26)
fit ELBO: -1.95 ± 0.03
fit distribution: Distributions.MvNormal{Float64, Pathfinder.WoodburyPDMat{Float64, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, Matrix{Float64}, Matrix{Float64}, Pathfinder.WoodburyPDFactorization{Float64, LinearAlgebra.Diagonal{Float64, Vector{Float64}}, LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}}, LinearAlgebra.UpperTriangular{Float64, Matrix{Float64}}}}, Vector{Float64}}(
dim: 3
μ: [1.4759842060455182, 2.0060275818274436, -1.5479825354725543]
Σ: [0.0018296810848421195 -0.0002589735319675424 -0.000618345292003903; -0.0002589735319675424 5.0682815503249e-5 5.534978328717582e-5; -0.000618345292003903 5.534978328717582e-5 0.0064256358711586746]
)
result_multi = multipathfinder(model, 1_000; nruns=n_chains)Multi-path Pathfinder result
runs: 8
draws: 1000
Pareto shape diagnostic: 0.49 (good)Here, the Pareto shape diagnostic indicates that it is likely safe to use these draws to compute posterior estimates.
When passed a Model, Pathfinder also gives access to the posterior draws in a familiar MCMCChains.Chains object.
result_multi.draws_transformedChains MCMC chain (1000×3×1 Array{Float64, 3}):
Iterations = 1:1:1000
Number of chains = 1
Samples per chain = 1000
parameters = α, β, σ
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rhat ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float64 ⋯
α 1.4748 0.0465 0.0015 931.3334 809.2502 1.0020 ⋯
β 2.0061 0.0076 0.0002 959.7284 825.6883 0.9999 ⋯
σ 0.2177 0.0151 0.0005 1008.8839 856.3196 0.9997 ⋯
1 column omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
α 1.3843 1.4443 1.4736 1.5070 1.5785
β 1.9906 2.0010 2.0064 2.0113 2.0214
σ 0.1867 0.2071 0.2175 0.2283 0.2487
We can also use these posterior draws to initialize MCMC sampling.
init_params = collect.(eachrow(result_multi.draws_transformed.value[1:n_chains, :, 1]))8-element Vector{Vector{Float64}}:
[1.4385712947204663, 2.0124531815275466, 0.22904948833216876]
[1.4310600350829619, 2.0147325585717835, 0.21996283990803556]
[1.5002556397980622, 2.003403475861116, 0.22333298780409003]
[1.4294167790341619, 2.01261352543737, 0.19601159618654096]
[1.4589565127911095, 2.0110309527354775, 0.20341109008509684]
[1.4852487222683175, 2.002876073142997, 0.23306041711870143]
[1.4594382602969214, 2.002946535777802, 0.21961434030119226]
[1.5219652756468702, 2.001773415039179, 0.19921379658345859]chns = sample(model, Turing.NUTS(), MCMCThreads(), 1_000, n_chains; init_params, progress=false)Chains MCMC chain (1000×15×8 Array{Float64, 3}):
Iterations = 501:1:1500
Number of chains = 8
Samples per chain = 1000
Wall duration = 6.59 seconds
Compute duration = 4.85 seconds
parameters = α, β, σ
internals = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rhat ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float64 ⋯
α 1.4767 0.0435 0.0008 3242.8022 3099.0731 1.0021 ⋯
β 2.0059 0.0075 0.0001 3463.0034 3814.4803 1.0016 ⋯
σ 0.2166 0.0157 0.0002 5169.4255 4494.8310 1.0025 ⋯
1 column omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
α 1.3918 1.4476 1.4769 1.5057 1.5627
β 1.9913 2.0009 2.0059 2.0110 2.0204
σ 0.1882 0.2055 0.2157 0.2267 0.2503
We can use Pathfinder's estimate of the metric and only perform enough warm-up to tune the step size.
inv_metric = result_multi.pathfinder_results[1].fit_distribution.Σ
metric = Pathfinder.RankUpdateEuclideanMetric(inv_metric)
kernel = HMCKernel(Trajectory{MultinomialTS}(Leapfrog(0.0), GeneralisedNoUTurn()))
adaptor = StepSizeAdaptor(0.8, 1.0) # adapt only the step size
nuts = AdvancedHMC.HMCSampler(kernel, metric, adaptor)
n_adapts = 50
n_draws = 1_000
chns = sample(
model,
externalsampler(nuts),
MCMCThreads(),
n_draws + n_adapts,
n_chains;
n_adapts,
init_params,
progress=false,
)[n_adapts + 1:end, :, :] # drop warm-up drawsChains MCMC chain (1000×16×8 Array{Float64, 3}):
Iterations = 51:1:1050
Number of chains = 8
Samples per chain = 1000
Wall duration = 2.75 seconds
Compute duration = 2.35 seconds
parameters = α, β, σ
internals = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size, is_adapt
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rhat ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float64 ⋯
α 1.4750 0.0424 0.0005 7523.8083 6310.0739 1.0004 ⋯
β 2.0060 0.0074 0.0001 6840.6333 5656.1131 1.0008 ⋯
σ 0.2168 0.0157 0.0002 6722.9557 5627.0576 1.0003 ⋯
1 column omitted
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
α 1.3916 1.4468 1.4750 1.5028 1.5585
β 1.9914 2.0011 2.0061 2.0110 2.0205
σ 0.1890 0.2057 0.2160 0.2267 0.2501
See Initializing HMC with Pathfinder for further examples.