AdvancedHMC.jl

struct ClassicNoUTurn{F<:AbstractFloat} <: AdvancedHMC.DynamicTerminationCriterion

Classic No-U-Turn criterion as described in Eq. (9) in [1].

Informally, this will terminate the trajectory expansion if continuing the simulation either forwards or backwards in time will decrease the distance between the left-most and right-most positions.

Fields

• max_depth::Int64

• Δ_max::AbstractFloat

References

1. 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)

HMCSampler

An AbstractMCMC.AbstractSampler for kernels in AdvancedHMC.jl.

Fields

• κ: AbstractMCMCKernel.

• metric: Choice of initial metric AbstractMetric. The metric type will be preserved during adaption.

• adaptor: AdvancedHMC.Adaptation.AbstractAdaptor.

Notes

Note that all the fields have the prefix initial_ to indicate that these will not necessarily correspond to the kernel, metric, and adaptor after sampling.

To access the updated fields, use the resulting HMCState.

HMC(ϵ::Real, n_leapfrog::Int)

Hamiltonian Monte Carlo sampler with static trajectory.

Fields

• n_leapfrog: Number of leapfrog steps.

• integrator: Choice of integrator, specified either using a Symbol or AbstractIntegrator

• metric: Choice of initial metric; Symbol means it is automatically initialised. The metric type will be preserved during automatic initialisation and adaption.

NUTS(δ::Real; max_depth::Int=10, Δ_max::Real=1000, integrator = :leapfrog, metric = :diagonal)

No-U-Turn Sampler (NUTS) sampler.

Fields

• δ: Target acceptance rate for dual averaging.

• max_depth: Maximum doubling tree depth.

• Δ_max: Maximum divergence during doubling tree.

• integrator: Choice of integrator, specified either using a Symbol or AbstractIntegrator

• metric: Choice of initial metric; Symbol means it is automatically initialised. The metric type will be preserved during automatic initialisation and adaption.

HMCDA(δ::Real, λ::Real, integrator = :leapfrog, metric = :diagonal)

Hamiltonian Monte Carlo sampler with Dual Averaging algorithm.

Fields

• δ: Target acceptance rate for dual averaging.

• λ: Target leapfrog length.

• integrator: Choice of integrator, specified either using a Symbol or AbstractIntegrator

• metric: Choice of initial metric; Symbol means it is automatically initialised. The metric type will be preserved during automatic initialisation and adaption.

Notes

For more information, please view the following paper (arXiv link):

• Hoffman, Matthew D., and Andrew Gelman. "The No-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo." Journal of Machine Learning Research 15, no. 1 (2014): 1593-1623.

sample([rng], a, [wv::AbstractWeights])

Select a single random element of a. Sampling probabilities are proportional to the weights given in wv, if provided.

Optionally specify a random number generator rng as the first argument (defaults to Random.default_rng()).

sample([rng], a, [wv::AbstractWeights], n::Integer; replace=true, ordered=false)

Select a random, optionally weighted sample of size n from an array a using a polyalgorithm. Sampling probabilities are proportional to the weights given in wv, if provided. replace dictates whether sampling is performed with replacement. ordered dictates whether an ordered sample (also called a sequential sample, i.e. a sample where items appear in the same order as in a) should be taken.

Optionally specify a random number generator rng as the first argument (defaults to Random.default_rng()).

sample([rng], a, [wv::AbstractWeights], dims::Dims; replace=true, ordered=false)

Select a random, optionally weighted sample from an array a specifying the dimensions dims of the output array. Sampling probabilities are proportional to the weights given in wv, if provided. replace dictates whether sampling is performed with replacement. ordered dictates whether an ordered sample (also called a sequential sample, i.e. a sample where items appear in the same order as in a) should be taken.

Optionally specify a random number generator rng as the first argument (defaults to Random.default_rng()).

sample([rng], wv::AbstractWeights)

Select a single random integer in 1:length(wv) with probabilities proportional to the weights given in wv.

Optionally specify a random number generator rng as the first argument (defaults to Random.default_rng()).

sample(
    rng::Random.AbatractRNG=Random.default_rng(),
    model::AbstractModel,
    sampler::AbstractSampler,
    N_or_isdone;
    kwargs...,
)

Sample from the model with the Markov chain Monte Carlo sampler and return the samples.

If N_or_isdone is an Integer, exactly N_or_isdone samples are returned.

Otherwise, sampling is performed until a convergence criterion N_or_isdone returns true. The convergence criterion has to be a function with the signature

isdone(rng, model, sampler, samples, state, iteration; kwargs...)

where state and iteration are the current state and iteration of the sampler, respectively. It should return true when sampling should end, and false otherwise.

sample(
    rng::Random.AbstractRNG=Random.default_rng(),
    model::AbstractModel,
    sampler::AbstractSampler,
    parallel::AbstractMCMCEnsemble,
    N::Integer,
    nchains::Integer;
    kwargs...,
)

Sample nchains Monte Carlo Markov chains from the model with the sampler in parallel using the parallel algorithm, and combine them into a single chain.

sample(
    rng::Random.AbstractRNG=Random.default_rng(),
    logdensity,
    sampler::AbstractSampler,
    N_or_isdone;
    kwargs...,
)

Wrap the logdensity function in a LogDensityModel, and call sample with the resulting model instead of logdensity.

The logdensity function has to support the LogDensityProblems.jl interface.

sample(
    rng::Random.AbstractRNG=Random.default_rng(),
    logdensity,
    sampler::AbstractSampler,
    parallel::AbstractMCMCEnsemble,
    N::Integer,
    nchains::Integer;
    kwargs...,
)

Wrap the logdensity function in a LogDensityModel, and call sample with the resulting model instead of logdensity.

The logdensity function has to support the LogDensityProblems.jl interface.

sample(
    rng::AbstractRNG,
    h::Hamiltonian,
    κ::AbstractMCMCKernel,
    θ::AbstractVecOrMat{T},
    n_samples::Int,
    adaptor::AbstractAdaptor=NoAdaptation(),
    n_adapts::Int=min(div(n_samples, 10), 1_000);
    drop_warmup::Bool=false,
    verbose::Bool=true,
    progress::Bool=false
)

Sample n_samples samples using the proposal κ under Hamiltonian h.

• The randomness is controlled by rng.
  • If rng is not provided, GLOBAL_RNG will be used.
• The initial point is given by θ.
• The adaptor is set by adaptor, for which the default is no adaptation.
  • It will perform n_adapts steps of adaptation, for which the default is the minimum of 1_000 and 10% of n_samples
• drop_warmup controls to drop the samples during adaptation phase or not
• verbose controls the verbosity
• progress controls whether to show the progress meter or not

abstract type AbstractIntegrator

Represents an integrator used to simulate the Hamiltonian system.

Implementation

A AbstractIntegrator is expected to have the following implementations:

• stat(@ref)
• nom_step_size(@ref)
• step_size(@ref)

abstract type AbstractMCMCKernel

Abstract type for HMC kernels.

abstract type AbstractMetric

Abstract type for preconditioning metrics.

abstract type AbstractTerminationCriterion

Abstract type for termination criteria for Hamiltonian trajectories, e.g. no-U-turn and fixed number of leapfrog integration steps.

How to sample a phase-point from the simulated trajectory.

A full binary tree trajectory with only necessary leaves and information stored.

struct ClassicNoUTurn{F<:AbstractFloat} <: AdvancedHMC.DynamicTerminationCriterion

Classic No-U-Turn criterion as described in Eq. (9) in [1].

Informally, this will terminate the trajectory expansion if continuing the simulation either forwards or backwards in time will decrease the distance between the left-most and right-most positions.

Fields

• max_depth::Int64

• Δ_max::AbstractFloat

References

1. 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)

abstract type DynamicTerminationCriterion <: AdvancedHMC.AbstractTerminationCriterion

Abstract type for dynamic Hamiltonian trajectory termination criteria.

Samples the end-point of the trajectory.

struct FixedIntegrationTime{F<:AbstractFloat} <: AdvancedHMC.StaticTerminationCriterion

Standard HMC implementation with a fixed integration time.

Fields

• λ::AbstractFloat: Total length of the trajectory, i.e. take floor(λ / integrator_step_size) number of leapfrog steps.

References

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

struct FixedNSteps <: AdvancedHMC.StaticTerminationCriterion

Static HMC with a fixed number of leapfrog steps.

Fields

• L::Int64: Number of steps to simulate, i.e. length of trajectory will be L + 1.

References

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

Completly resample new momentum.

struct GeneralisedNoUTurn{F<:AbstractFloat} <: AdvancedHMC.DynamicTerminationCriterion

Generalised No-U-Turn criterion as described in Section A.4.2 in [1].

Fields

• max_depth::Int64

• Δ_max::AbstractFloat

References

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

HMC(ϵ::Real, n_leapfrog::Int)

Hamiltonian Monte Carlo sampler with static trajectory.

Fields

• n_leapfrog: Number of leapfrog steps.

• integrator: Choice of integrator, specified either using a Symbol or AbstractIntegrator

• metric: Choice of initial metric; Symbol means it is automatically initialised. The metric type will be preserved during automatic initialisation and adaption.

HMCDA(δ::Real, λ::Real, integrator = :leapfrog, metric = :diagonal)

Hamiltonian Monte Carlo sampler with Dual Averaging algorithm.

Fields

• δ: Target acceptance rate for dual averaging.

• λ: Target leapfrog length.

• integrator: Choice of integrator, specified either using a Symbol or AbstractIntegrator

• metric: Choice of initial metric; Symbol means it is automatically initialised. The metric type will be preserved during automatic initialisation and adaption.

Notes

For more information, please view the following paper (arXiv link):

• Hoffman, Matthew D., and Andrew Gelman. "The No-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo." Journal of Machine Learning Research 15, no. 1 (2014): 1593-1623.

HMCProgressCallback

A callback to be used with AbstractMCMC.jl's interface, replicating the logging behavior of the non-AbstractMCMC sample.

Fields

• pm: Progress meter from ProgressMeters.jl.

• progress: Specifies whether or not to use display a progress bar.

• verbose: If progress is not specified and this is true some information will be logged upon completion of adaptation.

• num_divergent_transitions: Number of divergent transitions fo far.

• num_divergent_transitions_during_adaption

HMCSampler

An AbstractMCMC.AbstractSampler for kernels in AdvancedHMC.jl.

Fields

• κ: AbstractMCMCKernel.

• metric: Choice of initial metric AbstractMetric. The metric type will be preserved during adaption.

• adaptor: AdvancedHMC.Adaptation.AbstractAdaptor.

Notes

Note that all the fields have the prefix initial_ to indicate that these will not necessarily correspond to the kernel, metric, and adaptor after sampling.

To access the updated fields, use the resulting HMCState.

HMCState

Represents the state of a HMCSampler.

Fields

• i: Index of current iteration.

• transition: Current Transition.

• metric: Current AbstractMetric, possibly adapted.

• κ: Current AbstractMCMCKernel.

• adaptor: Current AbstractAdaptor.

struct JitteredLeapfrog{FT<:AbstractFloat, T<:Union{AbstractArray{FT<:AbstractFloat, 1}, FT<:AbstractFloat}} <: AdvancedHMC.AbstractLeapfrog{T<:Union{AbstractArray{FT<:AbstractFloat, 1}, FT<:AbstractFloat}}

Leapfrog integrator with randomly "jittered" step size ϵ for every trajectory.

Fields

• ϵ0::Union{AbstractVector{FT}, FT} where FT<:AbstractFloat: Nominal (non-jittered) step size.

• jitter::AbstractFloat: The proportion of the nominal step size ϵ0 that may be added or subtracted.

• ϵ::Union{AbstractVector{FT}, FT} where FT<:AbstractFloat: Current (jittered) step size.

Description

This is the same as LeapFrog(@ref) but with a "jittered" step size. This means that at the beginning of each trajectory we sample a step size ϵ by adding or subtracting from the nominal/base step size ϵ0 some random proportion of ϵ0, with the proportion specified by jitter, i.e. ϵ = ϵ0 - jitter * ϵ0 * rand(). p Jittering might help alleviate issues related to poor interactions with a fixed step size:

• In regions with high "curvature" the current choice of step size might mean over-shoot leading to almost all steps being rejected. Randomly sampling the step size at the beginning of the trajectories can therefore increase the probability of escaping such high-curvature regions.
• Exact periodicity of the simulated trajectories might occur, i.e. you might be so unlucky as to simulate the trajectory forwards in time L ϵ and ending up at the same point (which results in non-ergodicity; see Section 3.2 in [1]). If momentum is refreshed before each trajectory, then this should not happen exactly but it can still be an issue in practice. Randomly choosing the step-size ϵ might help alleviate such problems.

References

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

struct Leapfrog{T<:(Union{AbstractVector{var"#s29"}, var"#s29"} where var"#s29"<:AbstractFloat)} <: AdvancedHMC.AbstractLeapfrog{T<:(Union{AbstractVector{var"#s29"}, var"#s29"} where var"#s29"<:AbstractFloat)}

Leapfrog integrator with fixed step size ϵ.

Fields

• ϵ::Union{AbstractVector{var"#s29"}, var"#s29"} where var"#s29"<:AbstractFloat: Step size.

struct MultinomialTS{F<:AbstractFloat} <: AdvancedHMC.AbstractTrajectorySampler

Multinomial sampler for a trajectory consisting only a leaf node.

• tree weight is the (unnormalised) energy of the leaf.

struct MultinomialTS{F<:AbstractFloat} <: AdvancedHMC.AbstractTrajectorySampler

Multinomial sampler for the starting single leaf tree. (Log) weights for leaf nodes are their (unnormalised) Hamiltonian energies.

Ref: https://github.com/stan-dev/stan/blob/develop/src/stan/mcmc/hmc/nuts/base_nuts.hpp#L226

struct MultinomialTS{F<:AbstractFloat} <: AdvancedHMC.AbstractTrajectorySampler

Multinomial trajectory sampler carried during the building of the tree. It contains the weight of the tree, defined as the total probabilities of the leaves.

Fields

• zcand::AdvancedHMC.PhasePoint: Sampled candidate PhasePoint.

• ℓw::AbstractFloat: Total energy for the given tree, i.e. the sum of energies of all leaves.

NUTS(δ::Real; max_depth::Int=10, Δ_max::Real=1000, integrator = :leapfrog, metric = :diagonal)

No-U-Turn Sampler (NUTS) sampler.

Fields

• δ: Target acceptance rate for dual averaging.

• max_depth: Maximum doubling tree depth.

• Δ_max: Maximum divergence during doubling tree.

• integrator: Choice of integrator, specified either using a Symbol or AbstractIntegrator

• metric: Choice of initial metric; Symbol means it is automatically initialised. The metric type will be preserved during automatic initialisation and adaption.

struct PartialMomentumRefreshment{F<:AbstractFloat} <: AdvancedHMC.AbstractMomentumRefreshment

Partial momentum refreshment with refresh rate α.

Fields

• α::AbstractFloat

See equation (5.19) [1]

r' = α⋅r + sqrt(1-α²)⋅G

where r is the momentum and G is a Gaussian random variable.

References

1. Neal, Radford M. "MCMC using Hamiltonian dynamics." Handbook of markov chain monte carlo 2.11 (2011): 2.

struct SliceTS{F<:AbstractFloat} <: AdvancedHMC.AbstractTrajectorySampler

Slice sampler for the starting single leaf tree. Slice variable is initialized.

struct SliceTS{F<:AbstractFloat} <: AdvancedHMC.AbstractTrajectorySampler

Create a slice sampler for a single leaf tree:

• the slice variable is copied from the passed-in sampler s and
• the number of acceptable candicates is computed by comparing the slice variable against the current energy.

struct SliceTS{F<:AbstractFloat} <: AdvancedHMC.AbstractTrajectorySampler

Trajectory slice sampler carried during the building of the tree. It contains the slice variable and the number of acceptable condidates in the tree.

Fields

• zcand::AdvancedHMC.PhasePoint: Sampled candidate PhasePoint.

• ℓu::AbstractFloat: Slice variable in log-space.

• n::Int64: Number of acceptable candidates, i.e. those with probability larger than slice variable u.

abstract type StaticTerminationCriterion <: AdvancedHMC.AbstractTerminationCriterion

Abstract type for a fixed number of leapfrog integration steps.

struct StrictGeneralisedNoUTurn{F<:AbstractFloat} <: AdvancedHMC.DynamicTerminationCriterion

Generalised No-U-Turn criterion as described in Section A.4.2 in [1] with added U-turn check as described in [2].

Fields

• max_depth::Int64

• Δ_max::AbstractFloat

References

1. Betancourt, M. (2017). A Conceptual Introduction to Hamiltonian Monte Carlo. arXiv preprint arXiv:1701.02434.
2. https://github.com/stan-dev/stan/pull/2800

struct TemperedLeapfrog{FT<:AbstractFloat, T<:Union{AbstractArray{FT<:AbstractFloat, 1}, FT<:AbstractFloat}} <: AdvancedHMC.AbstractLeapfrog{T<:Union{AbstractArray{FT<:AbstractFloat, 1}, FT<:AbstractFloat}}

Tempered leapfrog integrator with fixed step size ϵ and "temperature" α.

Fields

• ϵ::Union{AbstractVector{FT}, FT} where FT<:AbstractFloat: Step size.

• α::AbstractFloat: Temperature parameter.

Description

Tempering can potentially allow greater exploration of the posterior, e.g. in a multi-modal posterior jumps between the modes can be more likely to occur.

Termination(s, nt, H0, H′)

Check termination of a Hamiltonian trajectory.

Termination(s, nt, H0, H′)

Check termination of a Hamiltonian trajectory.

Termination

Termination reasons

• dynamic: due to stoping criteria
• numerical: due to large energy deviation from starting (possibly numerical errors)

struct Trajectory{TS<:AdvancedHMC.AbstractTrajectorySampler, I<:AdvancedHMC.AbstractIntegrator, TC<:AdvancedHMC.AbstractTerminationCriterion}

Numerically simulated Hamiltonian trajectories.

struct Transition{P<:AdvancedHMC.PhasePoint, NT<:NamedTuple}

A transition that contains the phase point and other statistics of the transition.

Fields

• z::AdvancedHMC.PhasePoint: Phase-point for the transition.

• stat::NamedTuple: Statistics related to the transition, e.g. energy.

abstract type AbstractAdaptor

Abstract type for HMC adaptors.

An implementation of the Nesterov dual averaging algorithm to tune step size.

References

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. Nesterov, Y. (2009). Primal-dual subgradient methods for convex problems. Mathematical programming, 120(1), 221-259.