MCMC {adaptMCMC}  R Documentation 
(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 
n 
number of samples. 
init 
vector with initial values. 
scale 
vector with the variances or covariance matrix of the jump distribution. 
adapt 
if 
acc.rate 
desired acceptance rate (ignored if 
gamma 
controls the speed of adaption. Should be between 0.5 and 1. A lower gamma leads to faster adaption. 
list 
logical. If 
showProgressBar 
logical. If 
n.start 
iteration where the adaption starts. Only internally used. 
... 
further arguments passed to 
Details
The algorithm tunes the covariance matrix of the (normal) jump
distribution to achieve the desired acceptance rate. Classic
(nonadaptive) 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 
.
extra.values 
A list containing additional return values provided by

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), 9971008. doi:10.1007/s1122201192695.
See Also
MCMC.parallel
, MCMC.add.samples
Examples
## 
## Banana shaped distribution
## logpdf to sample from
p.log < function(x) {
B < 0.03 # controls 'bananacity'
x[1]^2/200  1/2*(x[2]+B*x[1]^2100*B)^2
}
## 
## generate samples
## 1) nonadaptive 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]^2100*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)