Multilevel Delayed Acceptance MCMC

.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. Un‐certain. 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.KeywordsMarkov chain Monte CarloBayesian inverse problemsmultilevel methodsmodel hierarchiesdetailed balancevariance reductionadaptive error modelMSC codes62F1562M0565C0565C40