# Multimodal distribution and Gelman-Rubin diagnostic

I recently implemented the Gelman-Rubin diagnostic for mcmc convergence and I was testing it with diferent posteriors to find the point where the chain is sampling the target distribution. To do this I find the first time where the Gelman-Rubin statistic differ from $1$ by a given tolerance (tol_default = .5).

The test work really well with Gaussian, Gaussian mixture and a mixture of Gaussian and uniform distributions but performs really poorly when I try to sample the posterior of the hyper parameters of a SE with noise gaussian process kernel.

Here we can see 6 chains sampling one of the 3 hyper parameters of my gaussian process and the Gelman-Rubin test give me a score = 2.6 but from the image I can see that the chain is stationary from time ~40000

Any sugestions?

• This question is rather unclear: the Gelman-Rubin intra/inter-variance criterion should be close to 1 to signal convergence. Second, your graph suggests the six chains are targetting different values. So they may be stationary but the stationary distributions obviously differ. It is thus no surprise the criterion cannot signal convergence! – Xi'an Mar 15 '17 at 7:57