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: 16)
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: [1.6508971085750253, 1.9311753921107158, 0.5801261338729266]
Σ: [0.11087798427694032 -0.016527256928018597 -0.0014208078651489035; -0.016527256928018624 0.0034071472308229607 0.00019433068465847872; -0.0014208078651489287 0.00019433068465846744 0.004718299969699233]
)
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: 0.39 (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 1.6542 0.3244 0.0104 976.4619 904.1276 1.0026 missing
β 1.9309 0.0586 0.0019 948.4244 966.2907 1.0007 missing
σ 1.8086 0.1246 0.0041 908.0277 944.1954 1.0031 missing
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
α 1.0317 1.4321 1.6640 1.8631 2.2388
β 1.8116 1.8937 1.9278 1.9706 2.0448
σ 1.6006 1.7206 1.8007 1.8855 2.0972We 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 = 0.05
┌ Info: Found initial step size
└ ϵ = 0.025
┌ Info: Found initial step size
└ ϵ = 0.05
┌ Info: Found initial step size
└ ϵ = 0.2
┌ Info: Found initial step size
└ ϵ = 0.05
┌ Info: Found initial step size
└ ϵ = 0.05
┌ Info: Found initial step size
└ ϵ = 0.025
Chains MCMC chain (1000×17×8 Array{Float64, 3}):
Iterations = 501:1:1500
Number of chains = 8
Samples per chain = 1000
Wall duration = 4.43 seconds
Compute duration = 2.98 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 1.6387 0.3304 0.0052 4009.3111 4441.1806 1.0035 1344.5040
β 1.9331 0.0580 0.0009 4049.6772 4048.3682 1.0032 1358.0406
σ 1.8170 0.1289 0.0019 4717.5282 4372.3596 1.0005 1582.0014
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
α 0.9950 1.4127 1.6399 1.8645 2.2844
β 1.8200 1.8927 1.9334 1.9723 2.0453
σ 1.5885 1.7267 1.8095 1.9006 2.0847We 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 1.6
[ Info: Found initial step size 3.2
[ Info: Found initial step size 3.2
[ Info: Found initial step size 3.2
[ Info: Found initial step size 1.6
[ Info: Found initial step size 3.2
[ Info: Found initial step size 1.7000000000000002
Chains MCMC chain (1000×17×8 Array{Float64, 3}):
Iterations = 51:1:1050
Number of chains = 8
Samples per chain = 1000
Wall duration = 1.41 seconds
Compute duration = 1.25 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 1.6446 0.3326 0.0032 10751.4061 6051.0525 1.0002 8580.5316
β 1.9315 0.0585 0.0006 10493.5935 6011.7122 1.0007 8374.7754
σ 1.8144 0.1273 0.0013 9918.0779 5971.6501 1.0011 7915.4652
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
α 0.9826 1.4201 1.6510 1.8688 2.3024
β 1.8160 1.8924 1.9317 1.9703 2.0472
σ 1.5886 1.7245 1.8056 1.8967 2.0824See Initializing HMC with Pathfinder for further examples.