Mixing of MCMC algorithms

ABSTRACT

We analyse MCMC chains focusing on how to find simulation parameters that give good mixing for discrete time, Harris ergodic Markov chains on a general state space X having invariant distribution π. The analysis uses an upper bound for the variance of the probability estimate. For each simulation parameter set, the bound is estimated from an MCMC chain using recurrence intervals. Recurrence intervals are a generalization of recurrence periods for discrete Markov chains. It is easy to compare the mixing properties for different simulation parameters. The paper gives general advice on how to improve the mixing of the MCMC chains and a new methodology for how to find an optimal acceptance rate for the Metropolis-Hastings algorithm. Several examples, both toy examples and large complex ones, illustrate how to apply the methodology in practice. We find that the optimal acceptance rate is smaller than the general recommendation in the literature in some of these examples.

KEYWORDS:

• Acceptance rates in Metropolis-Hastings algorithm
• MCMC
• mixing
• optimal step length
• recurrence
• variance estimates

Acknowledgement

The author thanks Marit Holden for testing the theory on the climate model, Gudmund Horn Hermansen for good advice on the presentation and Annabelle Redelmeier for carefully reading the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author.

ORCID

Lars Holden http://orcid.org/0000-0002-9475-5206

Additional information

Funding

This work was supported by Norges Forskningsråd [grant number Basic funding].