Random Walk Metropolis (RWM)

Random walk Metropolis-Hastings algorithm [44][61] in which parameters are sampled from symmetric distributions centered around the current values. The sampler simulates autocorrelated draws from a distribution that can be specified up to a constant of proportionality.

Model-Based Constructor

RWM(params::ElementOrVector{Symbol}, scale::ElementOrVector{T<:Real}; args...)

Construct a Sampler object for RWM sampling. Parameters are assumed to be continuous, but may be constrained or unconstrained.

Arguments

  • params : stochastic node(s) to be updated with the sampler. Constrained parameters are mapped to unconstrained space according to transformations defined by the Stochastic unlist() function.
  • scale : scaling value or vector of the same length as the combined elements of nodes params for the proposal distribution. Values are relative to the unconstrained parameter space, where candidate draws are generated.
  • args... : additional keyword arguments to be passed to the RWMVariate constructor.

Value

Returns a Sampler{RWMTune} type object.

Example

See the Dyes and other Examples.

Stand-Alone Function

sample!(v::RWMVariate)

Draw one sample from a target distribution using the RWM sampler. Parameters are assumed to be continuous and unconstrained.

Arguments

  • v : current state of parameters to be simulated.

Value

Returns v updated with simulated values and associated tuning parameters.

Example

The following example samples parameters in a simple linear regression model. Details of the model specification and posterior distribution can be found in the Supplement.

################################################################################
## Linear Regression
##   y ~ N(b0 + b1 * x, s2)
##   b0, b1 ~ N(0, 1000)
##   s2 ~ invgamma(0.001, 0.001)
################################################################################

using Mamba

## Data
data = Dict(
  :x => [1, 2, 3, 4, 5],
  :y => [1, 3, 3, 3, 5]
)

## Log-transformed Posterior(b0, b1, log(s2)) + Constant
logf = function(x::DenseVector)
   b0 = x[1]
   b1 = x[2]
   logs2 = x[3]
   r = data[:y] - b0 - b1 * data[:x]
   (-0.5 * length(data[:y]) - 0.001) * logs2 -
     (0.5 * dot(r, r) + 0.001) / exp(logs2) -
     0.5 * b0^2 / 1000 - 0.5 * b1^2 / 1000
end

## MCMC Simulation with Random Walk Metropolis
n = 5000
burnin = 1000
sim = Chains(n, 3, names = ["b0", "b1", "s2"])
theta = RWMVariate([0.0, 0.0, 0.0], [0.5, 0.25, 1.0], logf,
                   proposal = SymUniform)
for i in 1:n
  sample!(theta)
  sim[i, :, 1] = [theta[1:2]; exp(theta[3])]
end
describe(sim)

RWMVariate Type

Declaration

typealias RWMVariate SamplerVariate{RWMTune}

Fields

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

Constructor

RWMVariate(x::AbstractVector{T<:Real}, scale::ElementOrVector{U<:Real}, logf::Function; proposal::SymDistributionType=Normal)

Construct a RWMVariate object that stores simulated values and tuning parameters for RWM sampling.

Arguments

  • x : initial values.
  • scale : scalar or vector of the same length as x for the proposal distribution.
  • logf : function that takes a single DenseVector argument of parameter values at which to compute the log-transformed density (up to a normalizing constant).
  • proposal : symmetric distribution of type Biweight, Cosine, Epanechnikov, Normal, SymTriangularDist, SymUniform, or Triweight to be centered around current parameter values and used to generate proposal draws. Specified scale determines the standard deviations of Normal proposals and widths of the others.

Value

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

RWMTune Type

Declaration

type RWMTune <: SamplerTune

Fields

  • logf::Nullable{Function} : function supplied to the constructor to compute the log-transformed density, or null if not supplied.
  • scale::Union{Float64, Vector{Float64}} : scaling for the proposal distribution.
  • proposal::SymDistributionType : proposal distribution.