Nested Adaptation of MCMC Algorithms

. Markov chain Monte Carlo (MCMC) methods are ubiquitous tools for simulation-based inference in many fields but designing and identifying good MCMC samplers is still an open question. This paper introduces a novel MCMC algorithm, namely, Nested Adaptation MCMC. For sampling variables or blocks of variables, we use two levels of adaptation where the inner adaptation opti-mizes the MCMC performance within each sampler, while the outer adaptation explores the space of valid kernels to find the optimal samplers. We provide a theoretical foundation for our approach. To show the generality and usefulness of the approach, we describe a framework using only standard MCMC samplers as candidate samplers and some adaptation schemes for both inner and outer iterations. In several benchmark problems, we show that our proposed approach substantially outperforms other approaches, including an automatic blocking algorithm, in terms of MCMC efficiency and computational time.