Variance bounding and geometric ergodicity of Markov chain Monte Carlo kernels for approximate Bayesian computation

TL;DR

This paper investigates the variance bounding and geometric ergodicity of MCMC kernels in approximate Bayesian computation, highlighting issues with common kernels and proposing conditions under which a new kernel inherits desirable properties, with practical examples.

Contribution

It identifies limitations of existing MCMC kernels in ABC, and proves that a new kernel can inherit variance bounding and ergodicity under weak conditions, with cost bounds and illustrative examples.

Findings

Many common ABC MCMC kernels are not variance bounding or geometrically ergodic.

A new kernel can inherit these properties from the intractable Metropolis--Hastings kernel.

The computational cost of the new kernel is bounded under proper priors.

Abstract

Approximate Bayesian computation has emerged as a standard computational tool when dealing with the increasingly common scenario of completely intractable likelihood functions in Bayesian inference. We show that many common Markov chain Monte Carlo kernels used to facilitate inference in this setting can fail to be variance bounding, and hence geometrically ergodic, which can have consequences for the reliability of estimates in practice. This phenomenon is typically independent of the choice of tolerance in the approximation. We then prove that a recently introduced Markov kernel in this setting can inherit variance bounding and geometric ergodicity from its intractable Metropolis--Hastings counterpart, under reasonably weak and manageable conditions. We show that the computational cost of this alternative kernel is bounded whenever the prior is proper, and present indicative results on an example where spectral gaps and asymptotic variances can be computed, as well as an example involving inference for a partially and discretely observed, time-homogeneous, pure jump Markov process. We also supply two general theorems, one of which provides a simple sufficient condition for lack of variance bounding for reversible kernels and the other provides a positive result concerning inheritance of variance bounding and geometric ergodicity for mixtures of reversible kernels.