Adaptive Mixture Metropolis (AMM)¶
Implementation of the Roberts and Rosenthal [79] adaptive (multivariate) mixture Metropolis [45][46][63] sampler for simulating autocorrelated draws from a distribution that can be specified up to a constant of proportionality.
ModelBased Constructor¶

AMM
(params::ElementOrVector{Symbol}, Sigma::Matrix{T<:Real}; adapt::Symbol=:all, args...)¶ Construct a
Sampler
object for AMM 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 Stochasticunlist()
function.Sigma
: covariance matrix for the nonadaptive multivariate normal proposal distribution. The covariance matrix is relative to the unconstrained parameter space, where candidate draws are generated.adapt
: type of adaptation phase. Options are:all
: adapt proposal during all iterations.:burnin
: adapt proposal during burnin iterations.:none
: no adaptation (multivariate Metropolis sampling with fixed proposal).
args...
: additional keyword arguments to be passed to theAMMVariate
constructor.
Value
Returns aSampler{AMMTune}
type object.Example
StandAlone Function¶

sample!
(v::AMMVariate; adapt::Bool=true)¶ Draw one sample from a target distribution using the AMM sampler. Parameters are assumed to be continuous and unconstrained.
Arguments
v
: current state of parameters to be simulated. When running the sampler in adaptive mode, thev
argument in a successive call to the function will contain thetune
field returned by the previous call.adapt
: whether to adaptively update the proposal distribution.
Value
Returnsv
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] ) ## Logtransformed 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 Adaptive Multivariate Metopolis Sampling n = 5000 burnin = 1000 sim = Chains(n, 3, names = ["b0", "b1", "s2"]) theta = AMMVariate([0.0, 0.0, 0.0], eye(3), logf) for i in 1:n sample!(theta, adapt = (i <= burnin)) sim[i, :, 1] = [theta[1:2]; exp(theta[3])] end describe(sim)
AMMVariate Type¶
Declaration¶
const AMMVariate = SamplerVariate{AMMTune}
Fields¶
value::Vector{Float64}
: simulated values.tune::AMMTune
: tuning parameters for the sampling algorithm.
Constructor¶

AMMVariate
(x::AbstractVector{T<:Real}, Sigma::Matrix{U<:Real}, logf::Function; beta::Real=0.05, scale::Real=2.38)¶ Construct an
AMMVariate
object that stores simulated values and tuning parameters for AMM sampling.Arguments
x
: initial values.Sigma
: covariance matrix for the nonadaptive multivariate normal proposal distribution. The covariance matrix is relative to the unconstrained parameter space, where candidate draws are generated.logf
: function that takes a singleDenseVector
argument of parameter values at which to compute the logtransformed density (up to a normalizing constant).beta
: proportion of weight given to draws from the nonadaptive proposal with covariance factorizationSigmaL
, relative to draws from the adaptively tuned proposal with covariance factorizationSigmaLm
, during adaptive updating.scale
: factor (scale^2 / length(x)
) by which the adaptively updated covariance matrix is scaled—default value adopted from Gelman, Roberts, and Gilks [32].
Value
Returns anAMMVariate
type object with fields set to the suppliedx
and tuning parameter values.
AMMTune Type¶
Declaration¶
type AMMTune <: SamplerTune
Fields¶
logf::Nullable{Function}
: function supplied to the constructor to compute the logtransformed density, or null if not supplied.adapt::Bool
: whether the proposal distribution is being adaptively tuned.beta::Float64
: proportion of weight given to draws from the nonadaptive proposal with covariance factorizationSigmaL
, relative to draws from the adaptively tuned proposal with covariance factorizationSigmaLm
, during adaptive updating.m::Int
: number of adaptive update iterations that have been performed.Mv::Vector{Float64}
: running mean of drawsv
during adaptive updating. Used in the calculation ofSigmaLm
.Mvv::Matrix{Float64}
: running mean ofv * v'
during adaptive updating. Used in the calculation ofSigmaLm
.scale::Float64
: factor (scale^2 / length(v)
) by which the adaptively updated covariance matrix is scaled.SigmaL::LowerTriangular{Float64}
: Cholesky factorization of the nonadaptive covariance matrix.SigmaLm::Matrix{Float64}
: pivoted factorization of the adaptively tuned covariance matrix.