# Slice Simplex (SliceSimplex)¶

Implementation of the slice simplex sampler as described by Cowles et al. [19] for simulating autocorrelated draws of parameters on the simplex and from a distribution that can be specified up to a constant of proportionality.

## Model-Based Constructor¶

SliceSimplex(params::ElementOrVector{Symbol}; args...)

Construct a Sampler object for which SliceSimplex sampling is to be applied separately to each of the supplied parameters. Parameters are assumed to be continuous and constrained to a simplex.

Arguments

• params : stochastic node(s) to be updated with the sampler.
• args... : additional keyword arguments to be passed to the SliceSimplexVariate constructor.

Value

Returns a Sampler{SliceSimplexTune} type object.

Example

See the Asthma, Eyes, and other Examples.

## Stand-Alone Function¶

sample!(v::SliceSimplexVariate)

Draw one sample from a target distribution using the SliceSimplex sampler. Parameters are assumed to be continuous and constrained to a simplex.

Arguments

• v : current state of parameters to be simulated.

Value

Returns v updated with simulated values and associated tuning parameters.

Example

################################################################################
## Multinomial Model
##   y ~ Multinomial(n, rho)
##   rho ~ Dirichlet(1, ..., 1)
################################################################################

using Mamba

## Data
n, k = 100, 5
rho0 = rand(Dirichlet(ones(k)))
y = rand(Multinomial(n, rho0))

## Log-transformed Posterior(rho) + Constant
logf = function(rho::DenseVector)
logpdf(Multinomial(n, rho), y)
end

## MCMC Simulation with Slice Simplex Sampling
t = 10000
sim = Chains(t, k, names = map(i -> "rho[\$i]", 1:k))
rho = SliceSimplexVariate(fill(1 / k, k), logf)
for i in 1:t
sample!(rho)
sim[i, :, 1] = rho
end
describe(sim)

p = plot(sim)
draw(p, filename = "slicesimplexplot")


## SliceSimplexVariate Type¶

### Declaration¶

const SliceSimplexVariate = SamplerVariate{SliceSimplexTune}

### Fields¶

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

### Constructor¶

SliceSimplexVariate(x::AbstractVector{T<:Real}, logf::Function; scale::Real=1.0)

Construct a SliceSimplexVariate object that stores simulated values and tuning parameters for slice simplex sampling.

Arguments

• x : initial values.
• scale : value 0 < scale <= 1 by which to scale the standard simplex to define an initial space from which to simulate values.
• 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 SliceSimplexVariate type object with fields set to the supplied x and tuning parameter values.

## SliceSimplexTune Type¶

### Declaration¶

type SliceSimplexTune <: SamplerTune

### Fields¶

• logf::Nullable{Function} : function supplied to the constructor to compute the log-transformed density, or null if not supplied.
• scale::Float64 : value 0 < scale <= 1 by which to scale the standard simplex to define an initial space from which to simulate values.