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## Prerequisites

You should be familiar with the Metropolis–Hastings algorithm.

## Introduction

In this vignette we follow up on the original algorithm with a couple of points: numerical issues that were glossed over, and a useful plot.

## Avoiding numerical issues in acceptance probability

The key to the MH algorithm is computing the acceptance probability, which recall is given by $A= \min \left( 1, \frac{\pi(y)Q(x_t | y)}{\pi(x_t)Q(y | x_t)} \right).$

In practice both terms in the fraction may be very close to 0, so on a computer you should do this computation on the logarithmic scale before exponentiating; something like this: $\frac{\pi(y)Q(x_t | y)}{\pi(x_t)Q(y | x_t)} = \exp\left[ \log\pi(y) + \log Q(x_t | y) - \log \pi(x_t) - \log Q(y | x_t) \right]$

Furthermore, the log values in this expression should be computed directly, and not by computing them and then taking the log. For example, in our previous example we sampled from a target distribution that was the exponential: $\pi(x) = \exp(-x) \qquad (x>0)$ we we can directly compute $$\log \pi(x) = -x$$. The code we had in that previous example can therefore be written as follows:

log_target = function(x){
return(ifelse(x<0,-Inf,-x))
}

x = rep(0,10000)
x[1] = 100     #initialize; For purposes of illustration I set this to 100
for(i in 2:10000){
current_x = x[i-1]
proposed_x = current_x + rnorm(1,mean=0,sd=1)
A = exp(log_target(proposed_x) - log_target(current_x) )
if(runif(1)<A){
x[i] = proposed_x       # accept move with probabily min(1,A)
} else {
x[i] = current_x        # otherwise "reject" move, and stay where we are
}
}

## An important and useful plot

In practice MCMC is usually performed in high dimensional space. It can therefore be really hard to visualize directly the values of the chain. A simple 1-d summary that is always available is the log-target density, $$\log \pi(x)$$. So you should usually plot a trace of this whenever you run an MCMC scheme.

  plot(log_target(x), main = "log-target value")

Here the plot shows “typical” behaviour of MCMC scheme: because the starting point is chosen not to be close to the optimal of $$\pi$$, the chain initially takes some iterations to find a part of the space where $$\pi(x)$$ is large". Once it finds that part of the space it starts to explore around the region where $$\pi(x)$$ is large.

## Burn-in

The plot in the previous section immediately shows that there is an initial period of time where the Markov Chain is unduly influenced by its starting position. In other words, during those iterations the Markov Chain has not “converged” and those samples should not be considered to be samples from $$\pi$$. To address this it is common to discard the first set of iterations of any chain; the iterations that are discarded are often called “burn-in”.

Here, based on the plot we might discard the first 1000 iterations or so as burn-in. Here are comparisons of the samples with and without burnin discarded:

hist(x, main="without burn-in discarded")

hist(x[-(1:1000)], main="with burn-in discarded")

sessionInfo()
R version 4.1.0 Patched (2021-07-20 r80657)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Monterey 12.2

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.1-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.1-arm64/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base

loaded via a namespace (and not attached):
[1] Rcpp_1.0.7       whisker_0.4      knitr_1.36       magrittr_2.0.2
[5] workflowr_1.7.0  R6_2.5.1         rlang_0.4.12     fastmap_1.1.0
[9] fansi_0.5.0      highr_0.9        stringr_1.4.0    tools_4.1.0
[13] xfun_0.28        utf8_1.2.2       git2r_0.29.0     jquerylib_0.1.4
[17] htmltools_0.5.2  ellipsis_0.3.2   rprojroot_2.0.2  yaml_2.2.1
[21] digest_0.6.28    tibble_3.1.6     lifecycle_1.0.1  crayon_1.4.2
[25] later_1.3.0      sass_0.4.1       vctrs_0.3.8      fs_1.5.0
[29] promises_1.2.0.1 glue_1.5.0       evaluate_0.14    rmarkdown_2.11
[33] stringi_1.7.5    bslib_0.3.1      compiler_4.1.0   pillar_1.6.4
[37] jsonlite_1.7.2   httpuv_1.6.3     pkgconfig_2.0.3 

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