Running Pathfinder on Turing.jl models
This tutorial demonstrates how Turing can be used with Pathfinder.
We'll demonstrate with a regression example.
using AbstractMCMC, AdvancedHMC, DynamicPPL, FlexiChains, Pathfinder, Random, Turing
Random.seed!(39)
@model function regress(x)
α ~ Normal()
β ~ Normal()
σ ~ truncated(Normal(); lower=0)
μ = α .+ β .* x
y ~ product_distribution(Normal.(μ, σ))
end
x = 0:0.1:10
true_params = (; α=1.5, β=2, σ=2)
# simulate data
y = rand(regress(x) | true_params)[@varname(y)]
model = regress(x) | (; y)
n_chains = 8For convenience, pathfinder and multipathfinder can take Turing models as inputs and produce MCMCChains.Chains or FlexiChains.VNChain objects as outputs. To access this, we run Pathfinder normally; the chains representation of the draws (defaulting to Chains) is stored in draws_transformed.
result_single = pathfinder(model; ndraws=1_000)Single-path Pathfinder result
tries: 1
draws: 1000
fit iteration: 14 (total: 14)
fit ELBO: -213.66 ± 0.09
fit distribution: 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: [1.650897108575027, 1.9311753921107153, 0.5801261338729242]
Σ: [0.11087806908108015 -0.01652721016434848 -0.0014208266829752382; -0.016527210164348445 0.003407129429651901 0.0001943337674883388; -0.0014208266829752417 0.000194333767488339 0.00471830003589957]
)
result_single.draws_transformedChains MCMC chain (1000×6×1 Array{Float64, 3}):
Iterations = 1:1:1000
Number of chains = 1
Samples per chain = 1000
parameters = α, β, σ
internals = logprior, loglikelihood, logjoint
Use `describe(chains)` for summary statistics and quantiles.
To request a different chain type (e.g. VNChain), we can specify the chain_type directly.
pathfinder(model; ndraws=1_000, chain_type=VNChain).draws_transformedFlexiChain (1000 iterations, 1 chain)
↓ iter=1:1000 | → chain=1:1
Parameter type VarName
Parameters α, β, σ
Extra keys :logprior, :loglikelihood, :logjoint
Note that while Turing's sample methods default to initializing parameters from the prior with InitFromPrior, Pathfinder defaults to uniformly sampling them in the range [-2, 2] in unconstrained space (equivalent to Turing's InitFromUniform(-2, 2)). To use Turing's default in Pathfinder, specify init_sampler=InitFromPrior().
result_multi = multipathfinder(model, 1_000; nruns=n_chains, init_sampler=InitFromPrior())Multi-path Pathfinder result
runs: 8
draws: 1000
Pareto shape diagnostic: 0.25 (good)The Pareto shape diagnostic indicates that it is likely safe to use these draws to compute posterior estimates.
chns_pf = result_multi.draws_transformed
describe(chns_pf)Chains MCMC chain (1000×6×1 Array{Float64, 3}):
Iterations = 1:1:1000
Number of chains = 1
Samples per chain = 1000
parameters = α, β, σ
internals = logprior, loglikelihood, logjoint
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rhat ess_per_sec
Symbol Float64 Float64 Float64 Float64 Float64 Float64 Missing
α 1.6679 0.3343 0.0108 958.6483 993.2858 1.0026 missing
β 1.9288 0.0582 0.0019 919.4163 901.7198 1.0030 missing
σ 1.8103 0.1206 0.0039 937.8351 942.3972 0.9996 missing
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
α 0.9963 1.4330 1.6855 1.8904 2.3074
β 1.8174 1.8899 1.9284 1.9616 2.0491
σ 1.5897 1.7283 1.8111 1.8906 2.0518We can also use these draws to initialize MCMC sampling with InitFromParams.
params = AbstractMCMC.to_samples(DynamicPPL.ParamsWithStats, chns_pf[1:n_chains, :, :], model)
initial_params = [InitFromParams(p.params) for p in vec(params)]chns = sample(model, Turing.NUTS(), MCMCThreads(), 1_000, n_chains; initial_params, progress=false)
describe(chns)┌ Warning: Only a single thread available: MCMC chains are not sampled in parallel
└ @ AbstractMCMC ~/.julia/packages/AbstractMCMC/oqm6Y/src/sample.jl:544
┌ Info: Found initial step size
└ ϵ = 0.025
┌ Info: Found initial step size
└ ϵ = 0.0484375
┌ Info: Found initial step size
└ ϵ = 0.025
┌ Info: Found initial step size
└ ϵ = 0.0234375
┌ Info: Found initial step size
└ ϵ = 0.049218750000000006
┌ Info: Found initial step size
└ ϵ = 0.2
┌ Info: Found initial step size
└ ϵ = 0.025
┌ Info: Found initial step size
└ ϵ = 0.05
Chains MCMC chain (1000×17×8 Array{Float64, 3}):
Iterations = 501:1:1500
Number of chains = 8
Samples per chain = 1000
Wall duration = 4.08 seconds
Compute duration = 2.72 seconds
parameters = α, β, σ
internals = 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, logprior, loglikelihood, logjoint
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rhat ess_per_sec
Symbol Float64 Float64 Float64 Float64 Float64 Float64 Float64
α 1.6506 0.3406 0.0060 3180.4412 3591.6585 1.0028 1167.9916
β 1.9311 0.0605 0.0010 3313.3507 3921.0353 1.0018 1216.8016
σ 1.8163 0.1250 0.0018 4597.5509 4806.0162 1.0017 1688.4138
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
α 0.9812 1.4247 1.6505 1.8758 2.3321
β 1.8109 1.8912 1.9312 1.9722 2.0502
σ 1.5917 1.7288 1.8096 1.8951 2.0835We 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,
initial_params,
progress=false,
)[n_adapts + 1:end, :, :] # drop warm-up draws
describe(chns)┌ Warning: Only a single thread available: MCMC chains are not sampled in parallel
└ @ AbstractMCMC ~/.julia/packages/AbstractMCMC/oqm6Y/src/sample.jl:544
[ Info: Found initial step size 1.6
[ Info: Found initial step size 1.6500000000000001
[ Info: Found initial step size 1.6
[ Info: Found initial step size 3.2
[ Info: Found initial step size 1.6
[ Info: Found initial step size 1.6
[ Info: Found initial step size 1.6
[ Info: Found initial step size 1.6
Chains MCMC chain (1000×17×8 Array{Float64, 3}):
Iterations = 51:1:1050
Number of chains = 8
Samples per chain = 1000
Wall duration = 1.56 seconds
Compute duration = 1.39 seconds
parameters = α, β, σ
internals = 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, logprior, loglikelihood, logjoint
Summary Statistics
parameters mean std mcse ess_bulk ess_tail rhat ess_per_sec
Symbol Float64 Float64 Float64 Float64 Float64 Float64 Float64
α 1.6426 0.3416 0.0039 7880.5909 6002.8792 1.0003 5665.4140
β 1.9322 0.0595 0.0007 7963.7002 6011.4621 1.0005 5725.1619
σ 1.8139 0.1279 0.0016 6991.9770 5346.5114 1.0007 5026.5831
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
α 0.9726 1.4160 1.6391 1.8725 2.3134
β 1.8172 1.8920 1.9323 1.9721 2.0487
σ 1.5866 1.7261 1.8068 1.8917 2.0891See Initializing HMC with Pathfinder for further examples.