AdvancedHMC.jl

AdvancedHMC.jl provides a robust, modular, and efficient implementation of advanced HMC algorithms. An illustrative example of AdvancedHMC's usage is given below. AdvancedHMC.jl is part of Turing.jl, a probabilistic programming library in Julia. If you are interested in using AdvancedHMC.jl through a probabilistic programming language, please check it out!

Hands on AdvancedHMC.jl

Let's see how to sample a Hamiltonian using AdvanedHMC.jl

using AdvancedHMC, LogDensityProblems, ForwardDiff
# Define the target distribution using the `LogDensityProblem` interface
struct LogTargetDensity
    dim::Int
end
# standard multivariate normal distribution
LogDensityProblems.logdensity(p::LogTargetDensity, θ) = -sum(abs2, θ) / 2
LogDensityProblems.dimension(p::LogTargetDensity) = p.dim
function LogDensityProblems.capabilities(::Type{LogTargetDensity})
    return LogDensityProblems.LogDensityOrder{0}()
end

D = 10; # parameter dimensionality
initial_θ = rand(D); # initial parameter value
ℓπ = LogTargetDensity(D)

# Set the number of samples to draw and warmup iterations
n_samples, n_adapts = 2_000, 1_000

# Define a Hamiltonian system
metric = DiagEuclideanMetric(D)
hamiltonian = Hamiltonian(metric, ℓπ, ForwardDiff)

# Define a leapfrog solver, with the initial step size chosen heuristically
initial_ϵ = find_good_stepsize(hamiltonian, initial_θ)
integrator = Leapfrog(initial_ϵ)

# Define an HMC sampler with the following components
#   - multinomial sampling scheme,
#   - generalised No-U-Turn criteria, and
#   - windowed adaption for step-size and diagonal mass matrix
kernel = HMCKernel(Trajectory{MultinomialTS}(integrator, GeneralisedNoUTurn()))
adaptor = StanHMCAdaptor(MassMatrixAdaptor(metric), StepSizeAdaptor(0.8, integrator))

# Run the sampler to draw samples from the specified Gaussian, where
#   - `samples` will store the samples
#   - `stats` will store diagnostic statistics for each sample
samples, stats = sample(
    hamiltonian, kernel, initial_θ, n_samples, adaptor, n_adapts; progress=true
)

Citing AdvancedHMC.jl

If you use AdvancedHMC.jl for your own research, please consider citing the following publication:

Kai Xu, Hong Ge, Will Tebbutt, Mohamed Tarek, Martin Trapp, Zoubin Ghahramani: "AdvancedHMC.jl: A robust, modular and efficient implementation of advanced HMC algorithms.", Symposium on Advances in Approximate Bayesian Inference, 2020. (abs, pdf)

with the following BibTeX entry:

@inproceedings{xu2020advancedhmc,
  title={AdvancedHMC. jl: A robust, modular and efficient implementation of advanced HMC algorithms},
  author={Xu, Kai and Ge, Hong and Tebbutt, Will and Tarek, Mohamed and Trapp, Martin and Ghahramani, Zoubin},
  booktitle={Symposium on Advances in Approximate Bayesian Inference},
  pages={1--10},
  year={2020},
  organization={PMLR}
}

If you using AdvancedHMC.jl directly through Turing.jl, please consider citing the following publication:

Hong Ge, Kai Xu, and Zoubin Ghahramani: "Turing: a language for flexible probabilistic inference.", International Conference on Artificial Intelligence and Statistics, 2018. (abs, pdf)

with the following BibTeX entry:

@inproceedings{ge2018turing,
  title={Turing: A language for flexible probabilistic inference},
  author={Ge, Hong and Xu, Kai and Ghahramani, Zoubin},
  booktitle={International Conference on Artificial Intelligence and Statistics},
  pages={1682--1690},
  year={2018},
  organization={PMLR}
}

References

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

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. (arXiv)

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. (arXiv)