EMC (version 1.3)

MetropolisHastings: The Metropolis-Hastings algorithm

Description

Given a target density function and an asymmetric proposal distribution, this function produces samples from the target using the Metropolis Hastings algorithm.

Below sampDim refers to the dimension of the sample space.

Usage

MetropolisHastings(nIters, startingVal,          logTarDensFunc,       propNewFunc,          logPropDensFunc,      MHBlocks      = NULL, MHBlockNTimes = NULL, nThin         = 1, saveFitness   = FALSE,  verboseLevel  = 0, ...)

Arguments

nIters

integer $>$ 0.

startingVal

double vector of length sampDim.

logTarDensFunc

function of two arguments (draw, ...) that returns the target density evaluated in the log scale.

propNewFunc

function of three arguments (block, currentDraw, ...) that returns new Metropolis-Hastings proposals. See details below on the argument block.

logPropDensFunc

function of four arguments (block, currentDraw, proposalDraw, ...) that returns the proposal density evaluated in the log scale. See details below on the argument block.

MHBlocks

list of integer vectors giving dimensions to be blocked together for sampling. It defaults to as.list(1:sampDim), i.e., each dimension is treated as a block on its own. See details below for an example.

MHBlockNTimes

integer vector of number of times each block given by MHBlocks should be sampled in each iteration. It defaults to rep(1, length(MHBlocks)). See details below for an example.

nThin

integer $>=$ 1. Every nThin draw is saved.

saveFitness

logical indicating whether fitness values should be saved. See details below.

verboseLevel

integer, a value $>=$ 2 produces a lot of output.

...

optional arguments to be passed to logTarDensFunc, propNewFunc and logPropDensFunc.

Value

Below nSave refers to ceil(nIters / nThin). This function returns a list with the following components:

draws

matrix of dimension nSave $x$ sampDim, if saveFitness = FALSE. If saveFitness = TRUE, then the returned matrix is of dimension nSave $x$ (sampDim + 1), where the fitness values appear in its last column.

acceptRatios

matrix of the acceptance rates.

detailedAcceptRatios

matrix with detailed summary of the acceptance rates.

nIters

the nIters argument.

nThin

the nThin argument.

nSave

as defined above.

startingVal

the startingVal argument.

time

the time taken by the run.

Details

propNewFunc and logPropDensFunc

The propNewFunc and the logPropDensFunc are called multiple times by varying the block argument over 1:length(MHBlocks), so these functions should know how to generate a proposal from the currentDraw or to evaluate the proposal density depending on which block was passed as the argument. See the example section for sample code.

MHBlocks and MHBlockNTimes

Blocking is an important and useful tool in MCMC that helps speed up sampling and hence mixing. Example: Let sampDim = 6. Let we want to sample dimensions 1, 2, 4 as one block, dimensions 3 and 5 as another and treat dimension 6 as the third block. Suppose we want to sample the three blocks mentioned above 1, 5 and 10 times in each iteration, respectively. Then we could set MHBlocks = list(c(1, 2, 4), c(3, 5), 6) and MHBlockNTimes = c(1, 5, 10)

saveFitness

The term fitness refers to the negative of the logTarDensFunc values. By default, the fitness values are not saved, but one can do so by setting saveFitness = TRUE.

References

Jun S. Liu (2001). Monte Carlo strategies for scientific computing. Springer.

See Also

randomWalkMetropolis, parallelTempering, evolMonteCarlo

Examples

## Not run: 
# samplerObj <-
#     with(CigarShapedFuncGenerator2(-13579),
#          MetropolisHastings(nIters          = 5000,
#                             startingVal     = c(0, 0),
#                             logTarDensFunc  = logTarDensFunc,
#                             propNewFunc     = propNewFunc,
#                             logPropDensFunc = logPropDensFunc,
#                             verboseLevel    = 2))
# print(samplerObj)
# print(names(samplerObj))
# with(samplerObj,
#  {
#      print(detailedAcceptRatios)
#      print(dim(draws))
#      plot(draws,
#           xlim = c(-3, 5),
#           ylim = c(-3, 4),
#           pch  = '.',
#           ask  = FALSE,
#           main = as.expression(paste('# draws:', nIters)),
#           xlab = as.expression(substitute(x[xii], list(xii = 1))),
#           ylab = as.expression(substitute(x[xii], list(xii = 2))))    
#  })
# ## End(Not run)