# Quick illustration of Metropolis and Metropolis-in-Gibbs Sampling in R

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The code below gives a simple implementation of the Metropolis and Metropolis-in-Gibbs sampling algorithms, which are useful for sampling probability densities for which the normalizing constant is difficult to calculate, are irregular, or have high dimension (Metropolis-in-Gibbs).

## Metropolis sampling ## x - current value of Markov chain (numeric vector) ## targ - target log density function ## prop - function with prototype function(x, ...) that generates ## a proposal value from a symmetric proposal distribution library('mvtnorm') prop_mvnorm <- function(x, ...) drop(rmvnorm(1, mean=x, ...)) metropolis <- function(x, targ, prop=prop_mvnorm, ...) { xnew <- prop(x) lrat <- targ(xnew, ...) - targ(x, ...) if(log(runif(1)) < lrat) x <- xnew return(x) } ## Metropolis-in-Gibbs sampling ## x - current value of Markov chain (numeric vector) ## targ - target log density function ## ... - arguments passed to 'targ' gibbs <- function(x, targ, ...) { for(i in 1:length(x)) { ## define full conditional targ1 <- function(x1, ...) { x[i] <- x1 targ(x, ...) } ## sample using Metropolis algorithm x[i] <- metropolis(x[i], targ1, ...) } return(x) }

The following code produces the figure below to illustrate the two methods using a ‘dumbell’ distribution (cf. R package ‘ks’ vignette).

### The code below illustrates the use of the functions above ## target 'dumbell' density (c.f., R package 'ks' vignette) library('ks') mus <- rbind(c(-2,2), c(0,0), c(2,-2)) sigmas <- rbind(diag(2), matrix(c(0.8, -0.72, -0.72, 0.8), nrow=2), diag(2)) cwt <- 3/11 props <- c((1-cwt)/2, cwt, (1-cwt)/2) targ_test <- function(x, ...) log(dmvnorm.mixt(x, mus=mus, Sigmas=sigmas, props=props)) ## plot contours of target 'dumbell' density set.seed(42) par(mfrow=c(1,2)) plotmixt(mus=mus, Sigmas=sigmas, props=props, xlim=c(-4,4), ylim=c(-4,4), xlab=expression(x[1]), ylab=expression(x[2]), main="Metropolis-in-Gibbs") ## initialize and sample using Metropolis-in-Gibbs xcur <- gibbs(c(0,0), targ_test, sigma=vcov_test) for(j in 1:2000) { xcur <- gibbs(xcur, targ_test, sigma=vcov_test) points(xcur[1], xcur[2], pch=20, col='#00000055') } ## plot contours of target 'dumbell' density plotmixt(mus=mus, Sigmas=sigmas, props=props, xlim=c(-4,4), ylim=c(-4,4), xlab=expression(x[1]), ylab=expression(x[2]), main="Metropolis") ## initialize and sample using Metropolis xcur <- metropolis(c(0,0), targ_test, sigma=vcov_test) for(j in 1:2000) { xcur <- metropolis(xcur, targ_test, sigma=vcov_test) points(xcur[1], xcur[2], pch=20, col='#00000055') }

The figure illustrates two contrasting properties of the two methods:

- Metropolis-in-Gibbs samples can get ‘stuck’ in certain regions of the support, especially when there are multiple modes and/or significant correlation among the random variables. This is not as much a problem for Metropolis sampling.
- Metropolis sampling can produce fewer unique samples due to the poor approximation of the proposal density to the target density. This occurs more often for high-dimensional target densities.

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