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GitHub

AdvancedHMC.jl

Documentation for AdvancedHMC.jl

Types

AdvancedHMC.ClassicNoUTurn — Type
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)
source
AdvancedHMC.HMCSampler — Type
HMCSampler

An AbstractMCMC.AbstractSampler for kernels in AdvancedHMC.jl.

Fields

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.

source
AdvancedHMC.HMC — Type
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.

source
AdvancedHMC.NUTS — Type
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.

source
AdvancedHMC.HMCDA — Type
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.
source

Functions

StatsBase.sample — Function
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()).

source
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()).

source
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()).

source
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()).

source
sample(
     rng::Random.AbatractRNG=Random.default_rng(),
     model::AbstractModel,
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GitHub
AdvancedHMC.jl
CI DOI Coverage Status Stable Dev
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!
Interfaces
  • IMP.hmc: an experimental Python module for the Integrative Modeling Platform, which uses AdvancedHMC in its backend to sample protein structures.
NEWS
  • We presented a paper for AdvancedHMC.jl at AABI in 2019 in Vancouver, Canada. (abs, pdf, OpenReview)
  • We presented a poster for AdvancedHMC.jl at StanCon 2019 in Cambridge, UK. (pdf)
API CHANGES
  • [v0.5.0] Breaking! Convenience constructors for common samplers changed to:
    • HMC(leapfrog_stepsize::Real, n_leapfrog::Int)
    • NUTS(target_acceptance::Real) 
    • HMCDA(target_acceptance::Real, integration_time::Real)
  • [v0.2.22] Three functions are renamed.
    • Preconditioner(metric::AbstractMetric) -> MassMatrixAdaptor(metric) and 
    • NesterovDualAveraging(δ, integrator::AbstractIntegrator) -> StepSizeAdaptor(δ, integrator)
    • find_good_eps -> find_good_stepsize
  • [v0.2.15] n_adapts is no longer needed to construct StanHMCAdaptor; the old constructor is deprecated.
  • [v0.2.8] Two Hamiltonian trajectory sampling methods are renamed to avoid a name clash with Distributions.
    • Multinomial -> MultinomialTS
    • Slice -> SliceTS
  • [v0.2.0] The gradient function passed to Hamiltonian is supposed to return a value-gradient tuple now.
A minimal example - sampling from a multivariate Gaussian using NUTS
This section demonstrates a minimal example of sampling from a multivariate Gaussian (10-dimensional) using the no U-turn sampler (NUTS). Below we describe the major components of the Hamiltonian system which are essential to sample using this approach:
  • Metric: In many sampling problems the sample space is associated with a metric that allows us to measure the distance between any two points, and other similar quantities. In the example in this section, we use a special metric called the Euclidean Metric, represented with a D × D matrix from which we can compute distances.[1]
  • Leapfrog integration: Leapfrog integration is a second-order numerical method for integrating differential equations (In this case they are equations of motion for the relative position of one particle with respect to the other). The order of this integration signifies its rate of convergence. Any algorithm with a finite time step size will have numerical errors, and the order is related to this error. For a second-order algorithm, this error scales as the second power of the time step, hence, the name second-order. High-order integrators are usually complex to code and have a limited region of convergence; hence they do not allow arbitrarily large time steps. A second-order integrator is suitable for our purpose. Hence we opt for the leapfrog integrator. It is called leapfrog due to the ways this algorithm is written, where the positions and velocities of particles "leap over" each other.[2]
  • Kernel for trajectories (static or dynamic): Different kernels, which may be static or dynamic, can be used. At each iteration of any variant of the HMC algorithm, there are two main steps - the first step changes the momentum and the second step may change both the position and the momentum of a particle.[3]
using AdvancedHMC, ForwardDiff
 using LogDensityProblems
 using LinearAlgebra
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-AdvancedHMC.jl · AdvancedHMC
GitHub
AdvancedHMC.jl
Documentation for AdvancedHMC.jl
Types
AdvancedHMC.ClassicNoUTurn — Type
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)
source
AdvancedHMC.HMCSampler — Type
HMCSampler
An AbstractMCMC.AbstractSampler for kernels in AdvancedHMC.jl.
Fields
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.
source
AdvancedHMC.HMC — Type
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.
source
AdvancedHMC.NUTS — Type
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.
source
AdvancedHMC.HMCDA — Type
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.
source
Functions
StatsBase.sample — Function
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()).
source
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()).
source
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()).
source
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()).
source
sample(
+AdvancedHMC.jl · AdvancedHMC
+
+
+
+
+
+
GitHub
AdvancedHMC.jl
Documentation for AdvancedHMC.jl
Types
AdvancedHMC.ClassicNoUTurn — Type
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)
source
AdvancedHMC.HMCSampler — Type
HMCSampler
An AbstractMCMC.AbstractSampler for kernels in AdvancedHMC.jl.
Fields
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.
source
AdvancedHMC.HMC — Type
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.
source
AdvancedHMC.NUTS — Type
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.
source
AdvancedHMC.HMCDA — Type
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.
source
Functions
StatsBase.sample — Function
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()).
source
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()).
source
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()).
source
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()).
source
sample(
     rng::Random.AbatractRNG=Random.default_rng(),
     model::AbstractModel,
     sampler::AbstractSampler,
@@ -41,3 +499,4 @@
 )
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
source
More types
AdvancedHMC.AbstractIntegrator — Type
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)
source
AdvancedHMC.ClassicNoUTurn — Type
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)
source
AdvancedHMC.FixedIntegrationTime — Type
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) 
source
AdvancedHMC.FixedNSteps — Type
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)
source
AdvancedHMC.GeneralisedNoUTurn — Type
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.
source
AdvancedHMC.GeneralizedLeapfrog — Type
struct GeneralizedLeapfrog{T<:(Union{AbstractVector{var"#s48"}, var"#s48"} where var"#s48"<:AbstractFloat)} <: AdvancedHMC.AbstractLeapfrog{T<:(Union{AbstractVector{var"#s48"}, var"#s48"} where var"#s48"<:AbstractFloat)}
Generalized leapfrog integrator with fixed step size ϵ.
Fields
  • ϵ::Union{AbstractVector{var"#s48"}, var"#s48"} where var"#s48"<:AbstractFloat: Step size.
  • n::Int64
References
  1. Girolami, Mark, and Ben Calderhead. "Riemann manifold Langevin and Hamiltonian Monte Carlo methods." Journal of the Royal Statistical Society Series B: Statistical Methodology 73, no. 2 (2011): 123-214.
source
AdvancedHMC.HMC — Type
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.
source
AdvancedHMC.HMCDA — Type
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.
source
AdvancedHMC.HMCProgressCallback — Type
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
source
AdvancedHMC.HMCSampler — Type
HMCSampler
An AbstractMCMC.AbstractSampler for kernels in AdvancedHMC.jl.
Fields
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.
source
AdvancedHMC.JitteredLeapfrog — Type
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)
source
AdvancedHMC.Leapfrog — Type
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.
source
AdvancedHMC.MultinomialTS — Method
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.
source
AdvancedHMC.MultinomialTS — Method
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
source
AdvancedHMC.MultinomialTS — Type
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.
source
AdvancedHMC.NUTS — Type
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.
source
AdvancedHMC.PartialMomentumRefreshment — Type
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.
source
AdvancedHMC.SliceTS — Method
struct SliceTS{F<:AbstractFloat} <: AdvancedHMC.AbstractTrajectorySampler
Slice sampler for the starting single leaf tree. Slice variable is initialized.
source
AdvancedHMC.SliceTS — Method
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.
source
AdvancedHMC.SliceTS — Type
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.
source
AdvancedHMC.TemperedLeapfrog — Type
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.
source
AdvancedHMC.Termination — Type
Termination
Termination reasons
  • dynamic: due to stoping criteria
  • numerical: due to large energy deviation from starting (possibly numerical errors)
source
AdvancedHMC.Trajectory — Type
struct Trajectory{TS<:AdvancedHMC.AbstractTrajectorySampler, I<:AdvancedHMC.AbstractIntegrator, TC<:AdvancedHMC.AbstractTerminationCriterion}
Numerically simulated Hamiltonian trajectories.
source
AdvancedHMC.Transition — Type
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.
source
AdvancedHMC.Adaptation.NesterovDualAveraging — Type
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.
source

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diff --git a/v0.6.2/index.html b/v0.6.2/index.html
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-AdvancedHMC.jl · AdvancedHMC
GitHub
AdvancedHMC.jl
CI DOI Coverage Status Stable Dev
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!
Interfaces
  • IMP.hmc: an experimental Python module for the Integrative Modeling Platform, which uses AdvancedHMC in its backend to sample protein structures.
NEWS
  • We presented a paper for AdvancedHMC.jl at AABI in 2019 in Vancouver, Canada. (abs, pdf, OpenReview)
  • We presented a poster for AdvancedHMC.jl at StanCon 2019 in Cambridge, UK. (pdf)
API CHANGES
  • [v0.5.0] Breaking! Convenience constructors for common samplers changed to:
    • HMC(leapfrog_stepsize::Real, n_leapfrog::Int)
    • NUTS(target_acceptance::Real) 
    • HMCDA(target_acceptance::Real, integration_time::Real)
  • [v0.2.22] Three functions are renamed.
    • Preconditioner(metric::AbstractMetric) -> MassMatrixAdaptor(metric) and 
    • NesterovDualAveraging(δ, integrator::AbstractIntegrator) -> StepSizeAdaptor(δ, integrator)
    • find_good_eps -> find_good_stepsize
  • [v0.2.15] n_adapts is no longer needed to construct StanHMCAdaptor; the old constructor is deprecated.
  • [v0.2.8] Two Hamiltonian trajectory sampling methods are renamed to avoid a name clash with Distributions.
    • Multinomial -> MultinomialTS
    • Slice -> SliceTS
  • [v0.2.0] The gradient function passed to Hamiltonian is supposed to return a value-gradient tuple now.
A minimal example - sampling from a multivariate Gaussian using NUTS
This section demonstrates a minimal example of sampling from a multivariate Gaussian (10-dimensional) using the no U-turn sampler (NUTS). Below we describe the major components of the Hamiltonian system which are essential to sample using this approach:
  • Metric: In many sampling problems the sample space is associated with a metric that allows us to measure the distance between any two points, and other similar quantities. In the example in this section, we use a special metric called the Euclidean Metric, represented with a D × D matrix from which we can compute distances.[1]
  • Leapfrog integration: Leapfrog integration is a second-order numerical method for integrating differential equations (In this case they are equations of motion for the relative position of one particle with respect to the other). The order of this integration signifies its rate of convergence. Any algorithm with a finite time step size will have numerical errors, and the order is related to this error. For a second-order algorithm, this error scales as the second power of the time step, hence, the name second-order. High-order integrators are usually complex to code and have a limited region of convergence; hence they do not allow arbitrarily large time steps. A second-order integrator is suitable for our purpose. Hence we opt for the leapfrog integrator. It is called leapfrog due to the ways this algorithm is written, where the positions and velocities of particles "leap over" each other.[2]
  • Kernel for trajectories (static or dynamic): Different kernels, which may be static or dynamic, can be used. At each iteration of any variant of the HMC algorithm, there are two main steps - the first step changes the momentum and the second step may change both the position and the momentum of a particle.[3]
using AdvancedHMC, ForwardDiff
+AdvancedHMC.jl · AdvancedHMC
+
+
+
+
+
+
GitHub
AdvancedHMC.jl
CI DOI Coverage Status Stable Dev
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!
Interfaces
  • IMP.hmc: an experimental Python module for the Integrative Modeling Platform, which uses AdvancedHMC in its backend to sample protein structures.
NEWS
  • We presented a paper for AdvancedHMC.jl at AABI in 2019 in Vancouver, Canada. (abs, pdf, OpenReview)
  • We presented a poster for AdvancedHMC.jl at StanCon 2019 in Cambridge, UK. (pdf)
API CHANGES
  • [v0.5.0] Breaking! Convenience constructors for common samplers changed to:
    • HMC(leapfrog_stepsize::Real, n_leapfrog::Int)
    • NUTS(target_acceptance::Real) 
    • HMCDA(target_acceptance::Real, integration_time::Real)
  • [v0.2.22] Three functions are renamed.
    • Preconditioner(metric::AbstractMetric) -> MassMatrixAdaptor(metric) and 
    • NesterovDualAveraging(δ, integrator::AbstractIntegrator) -> StepSizeAdaptor(δ, integrator)
    • find_good_eps -> find_good_stepsize
  • [v0.2.15] n_adapts is no longer needed to construct StanHMCAdaptor; the old constructor is deprecated.
  • [v0.2.8] Two Hamiltonian trajectory sampling methods are renamed to avoid a name clash with Distributions.
    • Multinomial -> MultinomialTS
    • Slice -> SliceTS
  • [v0.2.0] The gradient function passed to Hamiltonian is supposed to return a value-gradient tuple now.
A minimal example - sampling from a multivariate Gaussian using NUTS
This section demonstrates a minimal example of sampling from a multivariate Gaussian (10-dimensional) using the no U-turn sampler (NUTS). Below we describe the major components of the Hamiltonian system which are essential to sample using this approach:
  • Metric: In many sampling problems the sample space is associated with a metric that allows us to measure the distance between any two points, and other similar quantities. In the example in this section, we use a special metric called the Euclidean Metric, represented with a D × D matrix from which we can compute distances.[1]
  • Leapfrog integration: Leapfrog integration is a second-order numerical method for integrating differential equations (In this case they are equations of motion for the relative position of one particle with respect to the other). The order of this integration signifies its rate of convergence. Any algorithm with a finite time step size will have numerical errors, and the order is related to this error. For a second-order algorithm, this error scales as the second power of the time step, hence, the name second-order. High-order integrators are usually complex to code and have a limited region of convergence; hence they do not allow arbitrarily large time steps. A second-order integrator is suitable for our purpose. Hence we opt for the leapfrog integrator. It is called leapfrog due to the ways this algorithm is written, where the positions and velocities of particles "leap over" each other.[2]
  • Kernel for trajectories (static or dynamic): Different kernels, which may be static or dynamic, can be used. At each iteration of any variant of the HMC algorithm, there are two main steps - the first step changes the momentum and the second step may change both the position and the momentum of a particle.[3]
using AdvancedHMC, ForwardDiff
 using LogDensityProblems
 using LinearAlgebra
 
@@ -132,3 +590,4 @@
   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)
Footnotes
  • 1The Euclidean metric is also known as the mass matrix in the physical perspective. See Hamiltonian mass matrix for available metrics.
  • 2About the leapfrog integration scheme: Suppose ${\bf x}$ and ${\bf v}$ are the position and velocity of an individual particle respectively; $i$ and $i+1$ are the indices for time values $t_i$ and $t_{i+1}$ respectively; $dt = t_{i+1} - t_i$ is the time step size (constant and regularly spaced intervals), and ${\bf a}$ is the acceleration induced on a particle by the forces of all other particles. Furthermore, suppose positions are defined at times $t_i, t_{i+1}, t_{i+2}, \dots $, spaced at constant intervals $dt$, the velocities are defined at halfway times in between, denoted by $t_{i-1/2}, t_{i+1/2}, t_{i+3/2}, \dots $, where $t_{i+1} - t_{i + 1/2} = t_{i + 1/2} - t_i = dt / 2$, and the accelerations ${\bf a}$ are defined only on integer times, just like the positions. Then the leapfrog integration scheme is given as: $x_{i} = x_{i-1} + v_{i-1/2} dt; \quad v_{i+1/2} = v_{i-1/2} + a_i dt$. For available integrators refer to Integrator.
  • 3On kernels: In the classical HMC approach, during the first step, new values for the momentum variables are randomly drawn from their Gaussian distribution, independently of the current values of the position variables. A Metropolis update is performed during the second step, using Hamiltonian dynamics to provide a new state. For available kernels refer to Kernel.

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