## Functions

# AdvancedHMC.AMethod.

A single Hamiltonian integration step.

NOTE: this function is intended to be used in find_good_stepsize only.

# AdvancedHMC.build_treeMethod.

Recursivly build a tree for a given depth j.

# AdvancedHMC.check_left_subtreeMethod.

check_left_subtree(h, t, tleft, tright)



Do a U-turn check between the leftmost phase point of t and the leftmost phase point of tright, the right subtree.

# AdvancedHMC.check_right_subtreeMethod.

Do a U-turn check between the rightmost phase point of t and the rightmost phase point of tleft, the left subtree.

# AdvancedHMC.combineMethod.

combine(treeleft, treeright)



Merge a left tree treeleft and a right tree treeright under given Hamiltonian h, then draw a new candidate sample and update related statistics for the resulting tree.

# AdvancedHMC.find_good_stepsizeMethod.

Find a good initial leap-frog step-size via heuristic search.

# AdvancedHMC.isterminatedMethod.

isterminated(_, h, t)



Detect U turn for two phase points (zleft and zright) under given Hamiltonian h using the (original) no-U-turn cirterion.

Ref: https://arxiv.org/abs/1111.4246, https://arxiv.org/abs/1701.02434

# AdvancedHMC.isterminatedMethod.

isterminated(_, h, t)



Detect U turn for two phase points (zleft and zright) under given Hamiltonian h using the generalised no-U-turn criterion.

Ref: https://arxiv.org/abs/1701.02434

# AdvancedHMC.isterminatedMethod.

isterminated(tc, h, t, tleft, tright)



Detect U turn for two phase points (zleft and zright) under given Hamiltonian h using the generalised no-U-turn criterion with additional U-turn checks.

Ref: https://arxiv.org/abs/1701.02434 https://github.com/stan-dev/stan/pull/2800

# AdvancedHMC.maxabsMethod.

maxabs(a, b)


Return the value with the largest absolute value.

# AdvancedHMC.mh_accept_ratioMethod.

Perform MH acceptance based on energy, i.e. negative log probability.

# AdvancedHMC.nom_step_sizeMethod.

nom_step_size(::AbstractIntegrator)


Get the nominal integration step size. The current integration step size may differ from this, for example if the step size is jittered. Nominal step size is usually used in adaptation.

# AdvancedHMC.pm_next!Method.

Progress meter update with all trajectory stats, iteration number and metric shown.

# AdvancedHMC.randcatMethod.

randcat(rng, P::AbstractMatrix)


Generating Categorical random variables in a vectorized mode. P is supposed to be a matrix of (D, N) where each column is a probability vector.

Example

P = [
0.5 0.3;
0.4 0.6;
0.1 0.1
]
u = [0.3, 0.4]
C = [
0.5 0.3
0.9 0.9
1.0 1.0
]


Then C .< u' is

[
0 1
0 0
0 0
]


thus convert.(Int, vec(sum(C .< u'; dims=1))) .+ 1 equals [1, 2].

# AdvancedHMC.simple_pm_next!Method.

Simple progress meter update without any show values.

# AdvancedHMC.statMethod.

Returns the statistics for transition t.

# AdvancedHMC.step_sizeFunction.

step_size(::AbstractIntegrator)


Get the current integration step size.

# AdvancedHMC.temperMethod.

temper(lf::TemperedLeapfrog, r, step::NamedTuple{(:i, :is_half),<:Tuple{Integer,Bool}}, n_steps::Int)


Tempering step. step is a named tuple with

• i being the current leapfrog iteration and
• is_half indicating whether or not it's (the first) half momentum/tempering step

# AdvancedHMC.transitionMethod.

transition(τ, h, z)



Make a MCMC transition from phase point z using the trajectory τ under Hamiltonian h.

NOTE: This is a RNG-implicit fallback function for transition(GLOBAL_RNG, τ, h, z)

# AdvancedHMC.update_nom_step_sizeFunction.

update_nom_step_size(i::AbstractIntegrator, ϵ) -> AbstractIntegrator


Return a copy of the integrator i with the new nominal step size (nom_step_size) ϵ.

# StatsBase.sampleMethod.

sample(
model::DifferentiableDensityModel,
N::Integer;
kwargs...
) -> Any



A convenient wrapper around AbstractMCMC.sample avoiding explicit construction of HMCSampler.

# StatsBase.sampleMethod.

sample(
rng::AbstractRNG,
h::Hamiltonian,
κ::AbstractMCMCKernel,
θ::AbstractVecOrMat{T},
n_samples::Int,
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

## Types

# AdvancedHMC.AbstractIntegratorType.

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)

# AdvancedHMC.AbstractTrajectorySamplerType.

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

# AdvancedHMC.BinaryTreeType.

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

# AdvancedHMC.ClassicNoUTurnType.

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)

# AdvancedHMC.DifferentiableDensityModelType.

DifferentiableDensityModel(ℓπ, ∂ℓπ∂θ)
DifferentiableDensityModel(ℓπ, m::Module)


A AbstractMCMC.AbstractMCMCModel representing a differentiable log-density.

If a module m is given as the second argument, then m is assumed to be an automatic-differentiation package and this will be used to compute the gradients.

Note that the module m must be imported before usage, e.g.

using Zygote: Zygote
model = DifferentiableDensityModel(ℓπ, Zygote)


results in a model which will use Zygote.jl as its AD-backend.

Fields

• ℓπ

: Log-density. Maps AbstractArray to value of the log-density.

• ∂ℓπ∂θ

: Gradient of log-density. Returns a tuple of ℓπ and the gradient evaluated at the given point.

# AdvancedHMC.EndPointTSType.

Samples the end-point of the trajectory.

# AdvancedHMC.FixedIntegrationTimeType.

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)

# AdvancedHMC.FixedNStepsType.

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)

# AdvancedHMC.FullMomentumRefreshmentType.

Completly resample new momentum.

# AdvancedHMC.GeneralisedNoUTurnType.

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.

# AdvancedHMC.HMCProgressCallbackType.

HMCProgressCallback


A callback to be used with AbstractMCMC.jl's interface, replicating the logging behavior of the non-AbstractMCMC [sample](/v0.22/docs/library/advancedhmc/#StatsBase.sample-Tuple{DifferentiableDensityModel, AdvancedHMC.AbstractMCMCKernel, AdvancedHMC.AbstractMetric, AdvancedHMC.Adaptation.AbstractAdaptor, Integer}).

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.

# AdvancedHMC.HMCSamplerType.

HMCSampler


A AbstractMCMC.AbstractSampler for kernels in AdvancedHMC.jl.

Fields

• initial_kernel

: Initial AbstractMCMCKernel.

• initial_metric

: Initial AbstractMetric.

• initial_adaptor

: Initial 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.

# AdvancedHMC.HMCStateType.

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.

# AdvancedHMC.JitteredLeapfrogType.

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)

# AdvancedHMC.LeapfrogType.

struct Leapfrog{T<:(Union{AbstractVector{var"#s67"}, var"#s67"} where var"#s67"<:AbstractFloat)} <: AdvancedHMC.AbstractLeapfrog{T<:(Union{AbstractVector{var"#s67"}, var"#s67"} where var"#s67"<:AbstractFloat)}


Leapfrog integrator with fixed step size ϵ.

Fields

• ϵ::Union{AbstractVector{var"#s67"}, var"#s67"} where var"#s67"<:AbstractFloat

: Step size.

# AdvancedHMC.MultinomialTSType.

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.

# AdvancedHMC.MultinomialTSMethod.

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.

# AdvancedHMC.MultinomialTSMethod.

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

# AdvancedHMC.NUTSMethod.

NUTS(int, args; kwargs...)



Convenient constructor for the no-U-turn sampler (NUTS). This falls back to HMCKernel(Trajectory{TS}(int, TC(args...; kwargs...))) where

• TS<:Union{MultinomialTS, SliceTS} is the type for trajectory sampler
• TC<:Union{ClassicNoUTurn, GeneralisedNoUTurn, StrictGeneralisedNoUTurn} is the type for termination criterion.

See ClassicNoUTurn, GeneralisedNoUTurn and StrictGeneralisedNoUTurn for details in parameters.

# AdvancedHMC.PartialMomentumRefreshmentType.

Partial momentum refreshment with refresh rate α.

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.

# AdvancedHMC.SliceTSType.

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.

# AdvancedHMC.SliceTSMethod.

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


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

# AdvancedHMC.SliceTSMethod.

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.

# AdvancedHMC.StrictGeneralisedNoUTurnType.

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

# AdvancedHMC.TemperedLeapfrogType.

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.

# AdvancedHMC.TerminationType.

Termination


Termination reasons

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

# AdvancedHMC.TerminationMethod.

Termination(s, nt, H0, H′)



Check termination of a Hamiltonian trajectory.

# AdvancedHMC.TerminationMethod.

Termination(s, nt, H0, H′)



Check termination of a Hamiltonian trajectory.

# AdvancedHMC.TrajectoryType.

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


Numerically simulated Hamiltonian trajectories.

# AdvancedHMC.TransitionType.

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.