Multilevel Delayed Acceptance MCMC

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. [SIAM/ASA J. Uncertain. Quantif., 3 (2015), pp. 1075-1108] in terms of the delayed acceptance MCMC of Christen and Fox [J. Comput. Graph. Statist., 14 (2005), pp. 795-810]. In particular, delayed acceptance 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 and 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.