Implementation of the Roberts and Rosenthal  adaptive (multivariate) mixture Metropolis  sampler for simulating autocorrelated draws from a distribution that can be specified up to a constant of proportionality.

## Stand-Alone Function¶

`amm!`(v::AMMVariate, SigmaF::Cholesky{Float64}, logf::Function; adapt::Bool=true)

Simulate one draw from a target distribution using an adaptive mixture Metropolis 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.
• `SigmaF` : Cholesky factorization of the covariance matrix for the non-adaptive multivariate normal proposal distribution.
• `logf` : function to compute the log-transformed density (up to a normalizing constant) at `v.value`.
• `adapt` : whether to adaptively update the proposal distribution.

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
b1 = x
logs2 = x
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])
SigmaF = cholfact(eye(3))
for i in 1:n
amm!(theta, SigmaF, logf, adapt = (i <= burnin))
sim[i,:,1] = [theta[1:2], exp(theta)]
end
describe(sim)
```

## AMMVariate Type¶

### Declaration¶

`AMMVariate <: VectorVariate`

### Fields¶

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

### Constructors¶

`AMMVariate`(x::Vector{VariateType}, tune::AMMTune)
`AMMVariate`(x::Vector{VariateType}, tune=nothing)

Construct a `AMMVariate` object that stores sampled values and tuning parameters for adaptive mixture Metropolis 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 `AMMVariate` type object with fields pointing to the values supplied to arguments `x` and `tune`.

## AMMTune Type¶

### Declaration¶

`type AMMTune`

### Fields¶

• `adapt::Bool` : whether the proposal distribution has been adaptively tuned.
• `beta::Real` : proportion of weight given to draws from the non-adaptive proposal with covariance factorization `SigmaF`, relative to draws from the adaptively tuned proposal with covariance factorization `SigmaLm`, during adaptive updating. Fixed at `beta = 0.05`.
• `m::Integer` : number of adaptive update iterations that have been performed.
• `Mv::Vector{Float64}` : running mean of draws `v` during adaptive updating. Used in the calculation of `SigmaLm`.
• `Mvv::Vector{Float64}` : running mean of `v * v'` during adaptive updating. Used in the calculation of `SigmaLm`.
• `scale::Real` : fixed value `2.38^2` in the factor (`scale / length(v)`) by which the adaptively updated covariance matrix is scaled—adopted from Gelman, Roberts, and Gilks .
• `SigmaF::Cholesky{Float64}` : factorization of the non-adaptive covariance matrix.
• `SigmaLm::Matrix{Float64}` : lower-triangular factorization of the adaptively tuned covariance matrix.

## Sampler Constructor¶

`AMM`(params::Vector{Symbol}, Sigma::Matrix{T<:Real}; adapt::Symbol=:all)

Construct a `Sampler` object for adaptive mixture Metropolis 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` : covariance matrix for the non-adaptive 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 burn-in iterations.
• `:none` : no adaptation (multivariate Metropolis sampling with fixed proposal).

Value

Returns a `Sampler` type object.

Example

See the Examples section.