/AdvancedHMC.jl

Efficient implementation of advanced Hamiltonian Monte Carlo algorithms

Primary LanguageJuliaMIT LicenseMIT

Efficient HMC implementations in Julia

Build Status Coverage Status

The code from this repository is used to implement HMC in Turing.jl. Try it out when it's available!

Minimal examples - sampling from a multivariate Gaussian using NUTS

using Distributions: MvNormal, logpdf
using ForwardDiff: gradient
using AdvancedHMC

# Define the target distribution and its gradient
const D = 10
const target = MvNormal(zeros(D), ones(D))
logπ(θ::AbstractVector{<:Real}) = logpdf(target, θ)
∂logπ∂θ(θ::AbstractVector{<:Real}) = gradient(logπ, θ)

# Sampling parameter settings
n_samples = 100_000
n_adapts = 2_000

# Initial points
θ_init = randn(D)

# Define metric space, Hamiltonian and sampling method
metric = DenseEuclideanMetric(D)
h = Hamiltonian(metric, logπ, ∂logπ∂θ)
prop = NUTS(Leapfrog(find_good_eps(h, θ_init)))
adaptor = StanNUTSAdaptor(n_adapts, PreConditioner(metric), NesterovDualAveraging(0.8, prop.integrator.ϵ))

# Sampling
samples = sample(h, prop, θ_init, n_samples, adaptor, n_adapts)

Reference

  1. Neal, R. M. (2011). MCMC using Hamiltonian dynamics. Handbook of Markov chain Monte Carlo, 2(11), 2. (pdf)

  2. Betancourt, M. (2017). A Conceptual Introduction to Hamiltonian Monte Carlo. arXiv preprint arXiv:1701.02434.

  3. Girolami, M., & Calderhead, B. (2011). Riemann manifold Langevin and Hamiltonian Monte Carlo methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73(2), 123-214. (link)

  4. Betancourt, M. J., Byrne, S., & Girolami, M. (2014). Optimizing the integrator step size for Hamiltonian Monte Carlo. arXiv preprint arXiv:1411.6669.

  5. Betancourt, M. (2016). Identifying the optimal integration time in Hamiltonian Monte Carlo. arXiv preprint arXiv:1601.00225.

  6. Hoffman, M. D., & Gelman, A. (2014). The No-U-Turn Sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. Journal of Machine Learning Research, 15(1), 1593-1623. (link)