Binary Hamiltonian Monte Carlo (BHMC)

Implementation of the binary-state Hamiltonian Monte Carlo sampler of Pakman [67]. The sampler simulates autocorrelated draws from a distribution that can be specified up to a constant of proportionality.

Model-Based Constructor

BHMC(params::ElementOrVector{Symbol}, traveltime::Real)

Construct a Sampler object for BHMC sampling. Parameters are assumed to have binary numerical values (0 or 1).

Arguments

• params : stochastic node(s) to be updated with the sampler.
• traveltime : length of time over which particle paths are simulated. It is recommended that supplied values be of the form , where optimal choices of are expected to grow with the parameter space dimensionality.

Value

Returns a Sampler{BHMCTune} type object.

Example

See the Pollution and other Examples.

Stand-Alone Function

sample!(v::BHMCVariate)

Draw one sample from a target distribution using the BHMC sampler. Parameters are assumed to have binary numerical values (0 or 1).

Arguments

• v : current state of parameters to be simulated.

Value

Returns v updated with simulated values and associated tuning parameters.

Example

################################################################################
## Linear Regression
##   y ~ MvNormal(X * (beta0 .* gamma), 1)
##   gamma ~ DiscreteUniform(0, 1)
################################################################################

using Mamba

## Data
n, p = 25, 10
X = randn(n, p)
beta0 = randn(p)
gamma0 = rand(0:1, p)
y = X * (beta0 .* gamma0) + randn(n)

## Log-transformed Posterior(gamma) + Constant
logf = function(gamma::DenseVector)
  logpdf(MvNormal(X * (beta0 .* gamma), 1.0), y)
end

## MCMC Simulation with Binary Hamiltonian Monte Carlo Sampling
t = 10000
sim = Chains(t, p, names = map(i -> "gamma[$i]", 1:p))
gamma = BHMCVariate(zeros(p), (2 * p + 0.5) * pi, logf)
for i in 1:t
  sample!(gamma)
  sim[i, :, 1] = gamma
end
describe(sim)

BHMCVariate Type

Declaration

typealias BHMCVariate SamplerVariate{BHMCTune}

Fields

• value::Vector{Float64} : simulated values.
• tune::BHMCTune : tuning parameters for the sampling algorithm.

Constructor

BHMCVariate(x::AbstractVector{T<:Real}, traveltime::Real, logf::Function)

Construct a BHMCVariate object that stores simulated values and tuning parameters for BHMC sampling.

Arguments

• x : initial values.
• traveltime : length of time over which particle paths are simulated. It is recommended that supplied values be of the form , where optimal choices of are expected to grow with the parameter space dimensionality.
• logf : function that takes a single DenseVector argument of parameter values at which to compute the log-transformed density (up to a normalizing constant).

Value

Returns a BHMCVariate type object with fields set to the supplied x and tuning parameter values.

BHMCTune Type

Declaration

type BHMCTune <: SamplerTune

Fields

• logf::Nullable{Function} : function supplied to the constructor to compute the log-transformed density, or null if not supplied.
• traveltime::Float64 : length of time over which particle paths are simulated.
• position::Vector{Float64} : initial particle positions.
• velocity::Vector{Float64} : initial particle velocites.
• wallhits::Int : number of times particles are reflected off the 0 threshold.
• wallcrosses::Int : number of times particles travel through the threshold.