Diagnosing issues in MCMC sampling

MCMC is your workhorse, so make sure it’s healthy

What’s MCMC

Well! I should write a well thought-out explanation here when it’s not late in the evening.

Relevant MCMC internals

This is my current understanding of which MCMC internals should be checked, how potential issues can be resolved, and what the internals reflect. No guarantee!

R hat

\(\hat{R}\) should be extremely close to 1 for all parameters.

\(\hat{R}\) (pronounced “R hat”) is a convergence measure (variance within vs between chains, I think) and must be extremely close to 1. Otherwise, our chains have not converged to the sample from the same posterior distribution. If \(\hat{R}\) is elevated, you can usually fix this by simply drawing more samples per chain. In case the posterior geometry can be simplified with, e.g., non centered parametrisation in hierarchical models, this is often a more efficient step to take.

ESS

Large effective sample size is important because it ensures stable estimates in the low-density regions of the posterior. Note that the effective sample size can be larger than the actual number of samples drawn.

Aim for ESS of at least 10 000 per parameter and posterior region (bulk / tail).

Be skeptical if your ESS bulk is much lower than the tail! You might be working with a multimodal posterior.

Divergent transitions

Something about skaters flying out into the infinite universe.

Divergent transitions are bad. We don’t want them.

If you have some, they should not be concentrated in any particular region of the posterior.

Tree depth

The tree depth can tell us if a model is poorly identified or (I think) has other sampling issues like a bimodal posterior.

If your tree depth hits the default maximum tree depth of 10, you’re in trouble.

If the tree depth is really high, the NUTS algorithm builds a very complicated sampling tree, which leads to very, very, very slow sampling.

Other MCMC internals

There are other internals like the leepfrog step size, loglikelihood (?), hamiltonian energy, energy error… not sure about these.