Generic adaptive Monte Carlo Markov Chain sampler

Description

Enables sampling from arbitrary distributions if the log density is known up to a constant; a common situation in the context of Bayesian inference. The implemented sampling algorithm was proposed by Vihola (2012) and achieves often a high efficiency by tuning the proposal distributions to a user defined acceptance rate.

Details

The workhorse function is MCMC. Chains can be updated with MCMC.add.samples. MCMC.parallel is a wrapper to generate independent chains on several CPU's in parallel using parallel. coda-functions can be used after conversion with convert.to.coda.

Author(s)

Andreas Scheidegger, andreas.scheidegger@eawag.ch or scheidegger.a@gmail.com

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, MCMC.add.samples, MCMC.parallel, convert.to.coda

(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

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:

.

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, andreas.scheidegger@eawag.ch or scheidegger.a@gmail.com.

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)

Add samples to an existing chain.

Description

Add samples to an existing chain produced by MCMC or MCMC.parallel.

Usage

MCMC.add.samples(MCMC.object, n.update, ...)

Arguments

Details

Only objects generated with the option list = TRUE can be updated.

A list of chains produced by MCMC.parallel can be updated. However, the calculations are not performed in parallel (i.e. only a single CPU is used).

Value

A updated version of MCMC.object.

Author(s)

Andreas Scheidegger, andreas.scheidegger@eawag.ch or scheidegger.a@gmail.com

See Also

MCMC, MCMC.parallel

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 200  samples

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


## ----------------------
## add 200 to the existing chain
samp <- MCMC.add.samples(samp, n.update=200)

str(samp)

Parallel computation of MCMC()

Description

A wrapper function to generate several independent Markov chains by stetting up cluster on a multi-core machine. The function is based on the parallel package.

Usage

MCMC.parallel(p, n, init, n.chain = 4, n.cpu, packages = NULL, dyn.libs=NULL,
    scale = rep(1, length(init)),  adapt = !is.null(acc.rate),
    acc.rate = NULL, gamma = 2/3, list = TRUE, ...)

Arguments

Details

This function is just a wrapper to use MCMC in parallel. It is based on parallel. Obviously, the application of this function makes only sense on a multi-core machine.

Value

A list with a list or matrix for each chain. See MCMC for details.

Author(s)

Andreas Scheidegger, andreas.scheidegger@eawag.ch or scheidegger.a@gmail.com

See Also

MCMC

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
## compute 4 independent chains on 2 CPU's (if available) in parallel

samp <- MCMC.parallel(p.log, n=200, init=c(x1=0, x2=1),
    n.chain=4, n.cpu=2, scale=c(1, 0.1),
    adapt=TRUE, acc.rate=0.234)

str(samp)

Converts chain(s) into coda objects.

Description

Converts chain(s) produced by MCMC or MCMC.parallel into coda objects.

Usage

convert.to.coda(sample)

Arguments

Details

Converts chain(s) produced by MCMC or MCMC.parallel so that they can be used with functions of the coda package.

Value

An object of the class mcmc or mcmc.list.

Author(s)

Andreas Scheidegger, andreas.scheidegger@eawag.ch or scheidegger.a@gmail.com

See Also

MCMC, mcmc, mcmc.list

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 200  samples

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


## ----------------------
## convert in object of class 'mcmc'
samp.coda <- convert.to.coda(samp)

class(samp.coda)

## ----------------------
## use functions of package 'coda'

require(coda)

plot(samp.coda)
cumuplot(samp.coda)