Gelman and Rubin Method for MCMC Convergence

The common problem when constructing a MCMC is when to stop it. It should be stopped when it converges to the target distribution. There are several purposed ideas but let's investigate the Gelman and Rubin Method.

The Gelman and Rubin Method (G&R) is based on multiple chains running at the same time. The starting points are chosen to be distant apart. When the characteristic of these chain i.e. the mean converge to about the same value, let's say it's about time to stop.

The idea of G&R method includes the between-sequence and within-sequence variances. The two variances is then used to calculated a factor, called "estimated potential scale reduction" $\hat R$.

The chains should continue to run if $\hat R$ is still high, until it is going below 1.1 or 1.2.

Example 14.9 of Computational Statistics Handbook with MATLAB shows the use of csgelrub() function for 4 MCMC (Metropolis-Hasting) chains. The characteristic is the mean of each chain. After the chain stopped, we can check the results as

>> nu(:,end)
ans =
    0.0248
   -0.0574
    0.0318
   -0.0196

>> rhat(end)
ans =
    1.0073