MCMC diagnostics for higher dimensions using Kullback Leibler divergence

ABSTRACT

In the existing literature of MCMC diagnostics, we have identified two areas for improvement. Firstly, the density-based diagnostic tools currently available in the literature are not equipped to assess the joint convergence of multiple variables. Secondly, in case of multi-modal target distribution if the MCMC sampler gets stuck in one of the modes, then the current diagnostic tools may falsely detect convergence. The Tool 1 proposed in this article makes use of adaptive kernel density estimation, symmetric Kullback Leibler divergence and a testing of hypothesis framework to assess the joint convergence of multiple variables. In cases where Tool 1 detects divergence of multiple chains, started at distinct initial values, we propose a visualization tool that can help to investigate reasons behind their divergence. The Tool 2 proposed in this article makes a novel use of the target distribution (known up till the unknown normalizing constant), to detect divergence when an MCMC sampler gets stuck in one of the modes of a multi-modal target distribution. The usefulness of the tools proposed in this article is illustrated using a multi-modal distribution, a mixture of bivariate normal distribution and a Bayesian logit model example.

KEYWORDS:

• Adaptive kernel density estimation
• convergence diagnostics
• Kullback Leibler divergence
• MCMC
• Monte Carlo

Acknowledgements

We would like to thank two reviewers, the AE and the Editor for their helpful comments that have improved the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.