Implementation of a Metropolis-within-Gibbs sampler [51][64][74] for iteratively simulating autocorrelated draws from a distribution that can be specified up to a constant of proportionality.

Stand-Alone Function¶

`amwg!`(v::AMWGVariate, sigma::Vector{Float64}, logf::Function; adapt::Bool=true, batchsize::Integer=50, target::Real=0.44)

Simulate one draw from a target distribution using an adaptive Metropolis-within-Gibbs 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, the `v` argument in a successive call to the function should contain the `tune` field returned by the previous call.
• `sigma` : initial standard deviations for the univariate normal proposal distributions.
• `logf` : function to compute the log-transformed density (up to a normalizing constant) at `v.value`.
• `adapt` : whether to adaptively update the proposal distribution.
• `batchsize` : number of samples that must be newly accumulated before applying an adaptive update to the proposal distributions.
• `target` : a target acceptance rate for the adaptive algorithm.

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 = [
:x => [1, 2, 3, 4, 5],
:y => [1, 3, 3, 3, 5]
]

## Log-transformed Posterior(b0, b1, log(s2)) + Constant
logf = function(x)
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 Metopolis-within-Gibbs Sampling
n = 5000
burnin = 1000
sim = Chains(n, 3, names = ["b0", "b1", "s2"])
theta = AMWGVariate([0.0, 0.0, 0.0])
sigma = ones(3)
for i in 1:n
amwg!(theta, sigma, logf, adapt = (i <= burnin))
sim[i,:,1] = [theta[1:2], exp(theta[3])]
end
describe(sim)
```

AMWGVariate Type¶

Declaration¶

`AMWGVariate <: VectorVariate`

Fields¶

• `value::Vector{VariateType}` : vector of sampled values.
• `tune::AMWGTune` : tuning parameters for the sampling algorithm.

Constructors¶

`AMWGVariate`(x::Vector{VariateType}, tune::AMWGTune)
`AMWGVariate`(x::Vector{VariateType}, tune=nothing)

Construct a `AMWGVariate` object that stores sampled values and tuning parameters for adaptive Metropolis-within-Gibbs sampling.

Arguments

• `x` : vector of sampled values.
• `tune` : tuning parameters for the sampling algorithm. If `nothing` is supplied, parameters are set to their defaults.

Value

Returns a `AMWGVariate` type object with fields pointing to the values supplied to arguments `x` and `tune`.

AMWGTune Type¶

Declaration¶

`type AMWGTune`

Fields¶

• `adapt::Bool` : whether the proposal distribution has been adaptively tuned.
• `accept::Vector{Integer}` : number of accepted candidate draws generated for each element of the parameter vector during adaptive updating.
• `batchsize::Integer` : number of samples that must be accumulated before applying an adaptive update to the proposal distributions.
• `m::Integer` : number of adaptive update iterations that have been performed.
• `sigma::Vector{Float64}` : updated values of the proposal standard deviations if `adapt = true`, and the user-defined values otherwise.
• `target::Real` : target acceptance rate for the adaptive algorithm.

Sampler Constructor¶

`AMWG`(params::Vector{Symbol}, sigma::Vector{T<:Real}; adapt::Symbol=:all, batchsize::Integer=50, target::Real=0.44)

Construct a `Sampler` object for adaptive Metropolis-within-Gibbs sampling. Parameters are assumed to be continuous, but may be constrained or unconstrained.

Arguments

• `params` : stochastic nodes to be updated with the sampler. Constrained parameters are mapped to unconstrained space according to transformations defined by the Stochastic `link()` function.

• `sigma` : initial standard deviations for the univariate normal proposal distributions. The standard deviations are relative to the unconstrained parameter space, where candidate draws are generated.

• `adapt` : type of adaptation phase. Options are
• `:all` : adapt proposals during all iterations.
• `:burnin` : adapt proposals during burn-in iterations.
• `:none` : no adaptation (Metropolis-within-Gibbs sampling with fixed proposals).
• `batchsize` : number of samples that must be accumulated before applying an adaptive update to the proposal distributions.

• `target` : a target acceptance rate for the algorithm.

Value

Returns a `Sampler` type object.

Example

See the Examples section.