Multilevel Delayed Acceptance MCMC

TL;DR

This paper introduces a multilevel delayed acceptance MCMC method that leverages a hierarchy of models to improve sampling efficiency for complex distributions, with proven ergodicity and variance reduction techniques.

Contribution

It extends the Delayed Acceptance MCMC framework to multiple model levels and subchains, providing a novel multilevel variance reduction and adaptive bias correction.

Findings

Demonstrated improved efficiency in Bayesian inverse problems

Proved ergodicity and detailed balance of the new algorithm

Provided software implementation in PyMC3

Abstract

We develop a novel Markov chain Monte Carlo (MCMC) method that exploits a hierarchy of models of increasing complexity to efficiently generate samples from an unnormalized target distribution. Broadly, the method rewrites the Multilevel MCMC approach of Dodwell et al. (2015) in terms of the Delayed Acceptance (DA) MCMC of Christen & Fox (2005). In particular, DA is extended to use a hierarchy of models of arbitrary depth, and allow subchains of arbitrary length. We show that the algorithm satisfies detailed balance, hence is ergodic for the target distribution. Furthermore, multilevel variance reduction is derived that exploits the multiple levels and subchains, and an adaptive multilevel correction to coarse-level biases is developed. Three numerical examples of Bayesian inverse problems are presented that demonstrate the advantages of these novel methods. The software and examples are available in PyMC3.