MCMCmetrop1R {MCMCpack} | R Documentation |
This function allows a user to construct a sample from a user-defined continuous distribution using a random walk Metropolis algorithm.
MCMCmetrop1R(fun, theta.init, burnin = 500, mcmc = 20000, thin = 1, tune = 1, verbose = 0, seed=NA, logfun = TRUE, force.samp = FALSE, V = NULL, optim.method = "BFGS", optim.lower = -Inf, optim.upper = Inf, optim.control = list(fnscale = -1, trace = 0, REPORT = 10, maxit=500), ...)
fun |
The unnormalized (log)density of the distribution from
which to take a sample. This must be a function defined in R whose first
argument is a continuous (possibly vector) variable. This first
argument is the point in the state space at which the (log)density
is to be evaluated. Additional arguments can be passed
to fun() by inserting them in the call to
MCMCmetrop1R() . See the Details section and the examples
below for more information. |
theta.init |
Starting values for the sampling. Must be of the
appropriate dimension. It must also be the case that
fun(theta.init, ...) is greater than -Inf if
fun() is a logdensity or greater than 0 if fun() is a
density. |
burnin |
The number of burn-in iterations for the sampler. |
mcmc |
The number of MCMC iterations after burnin. |
thin |
The thinning interval used in the simulation. The number of MCMC iterations must be divisible by this value. |
tune |
The tuning parameter for the Metropolis sampling. Can be either a positive scalar or a k-vector, where k is the length of theta. |
verbose |
A switch which determines whether or not the progress of
the sampler is printed to the screen. If verbose is greater
than 0 the iteration number, the
theta vector, the function value, and the Metropolis
acceptance rate are sent to the screen every verbose th iteration. |
seed |
The seed for the random number generator. If NA, the Mersenne
Twister generator is used with default seed 12345; if an integer is
passed it is used to seed the Mersenne twister. The user can also
pass a list of length two to use the L'Ecuyer random number generator,
which is suitable for parallel computation. The first element of the
list is the L'Ecuyer seed, which is a vector of length six or NA (if NA
a default seed of rep(12345,6) is used). The second element of
list is a positive substream number. See the MCMCpack
specification for more details. |
logfun |
Logical indicating whether fun returns the natural log
of a density function (TRUE) or a density (FALSE). |
force.samp |
Logical indicating whether the sampling should
proceed if the Hessian calculated from the initial call to
optim routine to maximize the (log)density
is not negative definite. If force.samp==TRUE
and the Hessian from optim is non-negative definite, the
Hessian is rescaled by subtracting small values from it's main
diagonal until it is negative definite. Sampling proceeds using
this rescaled Hessian in place of the original Hessian from
optim . By default, if force.samp==FALSE and the Hessian from
optim is non-negative definite, an error message is
printed and the call to MCMCmetrop1R is terminated.
Please note that a non-negative Hessian at the mode is often an indication that the function of interest is not a proper density. Thus, force.samp should only be set equal to
TRUE with great caution.
|
V |
The variance-covariance matrix for the Gaussian proposal
distribution. Must be a square matrix or NULL . If a square
matrix, V must have dimension equal to the length of
theta.init . If NULL , V is calculated from
tune and an initial call to optim . See the Details
section below for more information. Unless the log-posterior is
expensive to compute it will typically be best to use the default
V = NULL . |
optim.method |
The value of the method parameter sent to
optim during an initial maximization of fun . See
optim for more details. |
optim.lower |
The value of the lower parameter sent to
optim during an initial maximization of fun . See
optim for more details. |
optim.upper |
The value of the upper parameter sent to
optim during an initial maximization of fun . See
optim for more details. |
optim.control |
The value of the control parameter sent to
optim during an initial maximization of fun . See
optim for more details. |
... |
Additional arguments. |
MCMCmetrop1R produces a sample from a user-defined distribution using a random walk Metropolis algorithm with multivariate normal proposal distribution. See Gelman et al. (2003) and Robert & Casella (2004) for details of the random walk Metropolis algorithm.
The proposal distribution is centered at the current value of
theta and has variance-covariance V. If
V is specified by the user to be NULL
then V is
calculated as: V = T (-1*H)^{-1} T, where T is a the
diagonal positive definite matrix formed from the tune
and
H is the approximate Hessian of fun
evaluated at its
mode. This last calculation is done via an initial call to
optim
.
An mcmc object that contains the posterior sample. This object can be summarized by functions provided by the coda package.
Andrew Gelman, John B. Carlin, Hal S. Stern, and Donald B. Rubin. 2003. Bayesian Data Analysis. 2nd Edition. Boca Raton: Chapman & Hall/CRC.
Andrew D. Martin, Kevin M. Quinn, and Daniel Pemstein. 2004. Scythe Statistical Library 1.0. http://scythe.wustl.edu.
Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2002. Output Analysis and Diagnostics for MCMC (CODA). http://www-fis.iarc.fr/coda/.
Christian P. Robert and George Casella. 2004. Monte Carlo Statistical Methods. 2nd Edition. New York: Springer.
plot.mcmc
,
summary.mcmc
, optim
,
metrop
## Not run: ## logistic regression with an improper uniform prior ## X and y are passed as args to MCMCmetrop1R logitfun <- function(beta, y, X){ eta <- X %*% beta p <- 1.0/(1.0+exp(-eta)) sum( y * log(p) + (1-y)*log(1-p) ) } x1 <- rnorm(1000) x2 <- rnorm(1000) Xdata <- cbind(1,x1,x2) p <- exp(.5 - x1 + x2)/(1+exp(.5 - x1 + x2)) yvector <- rbinom(1000, 1, p) post.samp <- MCMCmetrop1R(logitfun, theta.init=c(0,0,0), X=Xdata, y=yvector, thin=1, mcmc=40000, burnin=500, tune=c(1.5, 1.5, 1.5), verbose=500, logfun=TRUE) raftery.diag(post.samp) plot(post.samp) summary(post.samp) ## ################################################## ## negative binomial regression with an improper unform prior ## X and y are passed as args to MCMCmetrop1R negbinfun <- function(theta, y, X){ k <- length(theta) beta <- theta[1:(k-1)] alpha <- exp(theta[k]) mu <- exp(X %*% beta) log.like <- sum( lgamma(y+alpha) - lfactorial(y) - lgamma(alpha) + alpha * log(alpha/(alpha+mu)) + y * log(mu/(alpha+mu)) ) } n <- 1000 x1 <- rnorm(n) x2 <- rnorm(n) XX <- cbind(1,x1,x2) mu <- exp(1.5+x1+2*x2)*rgamma(n,1) yy <- rpois(n, mu) post.samp <- MCMCmetrop1R(negbinfun, theta.init=c(0,0,0,0), y=yy, X=XX, thin=1, mcmc=35000, burnin=1000, tune=1.5, verbose=500, logfun=TRUE, seed=list(NA,1)) raftery.diag(post.samp) plot(post.samp) summary(post.samp) ## ################################################## ## sample from a univariate normal distribution with ## mean 5 and standard deviation 0.1 ## ## (MCMC obviously not necessary here and this should ## really be done with the logdensity for better ## numerical accuracy-- this is just an illustration of how ## MCMCmetrop1R works with a density rather than logdensity) post.samp <- MCMCmetrop1R(dnorm, theta.init=5.3, mean=5, sd=0.1, thin=1, mcmc=50000, burnin=500, tune=2.0, verbose=5000, logfun=FALSE) summary(post.samp) ## End(Not run)