MCMC {adaptMCMC}

(Adaptive) Metropolis Sampler

Description

Implementation of the robust adaptive Metropolis sampler of Vihola (2012).

Usage

MCMC(p, n, init, scale = rep(1, length(init)),
    adapt = !is.null(acc.rate), acc.rate = NULL, gamma = 2/3,
    list = TRUE, showProgressBar=interactive(), n.start = 0, ...)

Arguments

p

function that returns a value proportional to the log probability density to sample from. Alternatively it can be a function that returns a list with at least one element named log.density. See details below.

n

number of samples.

init

vector with initial values.

scale

vector with the variances or covariance matrix of the jump distribution.

adapt

if TRUE, adaptive sampling is used, if FALSE classic metropolis sampling, if a positive integer the adaption stops after adapt iterations.

acc.rate

desired acceptance rate (ignored if adapt=FALSE)

gamma

controls the speed of adaption. Should be between 0.5 and 1. A lower gamma leads to faster adaption.

list

logical. If TRUE a list is returned otherwise only a matrix with the samples.

showProgressBar

logical. If TRUE a progress bar is shown.

n.start

iteration where the adaption starts. Only internally used.

...

further arguments passed to p.

Details

The algorithm tunes the covariance matrix of the (normal) jump distribution to achieve the desired acceptance rate. Classic (non-adaptive) Metropolis sampling can be obtained by setting adapt=FALSE.

Note, due to the calculation for the adaption steps the sampler is rather slow. However, with a suitable jump distribution good mixing can be observed with less samples. This is crucial if the computation of p is slow.

In some cases the function p may not only calculate the log density but return a list containing also other values. For example if p is a log posterior one may be also interested to store the corresponding prior and likelihood values. The function must either return always a scalar or always a list, however, the length of the list may vary.

Value

If list=FALSE a matrix is with the samples.

If list=TRUE a list is returned with the following components:

samples

matrix with samples

log.p

vector with the (unnormalized) log density for each sample

n.sample

number of generated samples

acceptance.rate

acceptance rate

adaption

either logical if adaption was used or not, or the number of adaption steps.

sampling.parameters

a list with further sampling parameters. Mainly used by MCMC.add.samples()

.

extra.values

A list containing additional return values provided by p. Only if p provides a list.

Note

Due to numerical errors it may happen that the computed covariance matrix is not positive definite. In such a case the nearest positive definite matrix is calculated with nearPD() from the package Matrix.

Author(s)

Andreas Scheidegger, [e-mail] or [e-mail].

Thanks to David Pleydell, Venelin, and Umberto Picchini for spotting errors and providing improvements. Ian Taylor implemented the usage of adapt_S which lead to a nice speedup.

References

Vihola, M. (2012) Robust adaptive Metropolis algorithm with coerced acceptance rate. Statistics and Computing, 22(5), 997-1008. doi:10.1007/s11222-011-9269-5.

See Also

MCMC.parallel, MCMC.add.samples

Examples

## ----------------------
## Banana shaped distribution

## log-pdf to sample from
p.log <- function(x) {
  B <- 0.03                              # controls 'bananacity'
  -x[1]^2/200 - 1/2*(x[2]+B*x[1]^2-100*B)^2
}


## ----------------------
## generate samples

## 1) non-adaptive sampling
samp.1 <- MCMC(p.log, n=200, init=c(0, 1), scale=c(1, 0.1),
               adapt=FALSE)

## 2) adaptive sampling
samp.2 <- MCMC(p.log, n=200, init=c(0, 1), scale=c(1, 0.1),
               adapt=TRUE, acc.rate=0.234)


## ----------------------
## summarize results

str(samp.2)
summary(samp.2$samples)

## covariance of last jump distribution
samp.2$cov.jump


## ----------------------
## plot density and samples

x1 <- seq(-15, 15, length=80)
x2 <- seq(-15, 15, length=80)
d.banana <- matrix(apply(expand.grid(x1, x2), 1,  p.log), nrow=80)

par(mfrow=c(1,2))
image(x1, x2, exp(d.banana), col=cm.colors(60), asp=1, main="no adaption")
contour(x1, x2, exp(d.banana), add=TRUE, col=gray(0.6))
lines(samp.1$samples, type='b', pch=3)

image(x1, x2, exp(d.banana), col=cm.colors(60), asp=1, main="with adaption")
contour(x1, x2, exp(d.banana), add=TRUE, col=gray(0.6))
lines(samp.2$samples, type='b', pch=3)



## ----------------------
## function returning extra information in a list


p.log.list <- function(x) {
  B <- 0.03                              # controls 'bananacity'
  log.density <- -x[1]^2/200 - 1/2*(x[2]+B*x[1]^2-100*B)^2

  result <- list(log.density=log.density)

  ## under some conditions one may want to return other infos
  if(x[1]<0) {
    result$message <- "Attention x[1] is negative!"
    result$x <- x[1]
  }

  result
}

samp.list <- MCMC(p.log.list, n=200, init=c(0, 1), scale=c(1, 0.1),
                  adapt=TRUE, acc.rate=0.234)

## the additional values are stored under `extras.values`
head(samp.list$extras.values)