Mixing Regimes for Adaptive Markov Chain Monte Carlo

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Calculating complicated integrals is a central computational problem in Bayesian statistics and the sciences. Markov chain Monte Carlo (MCMC) is a general-purpose tool for doing this computation by generating a sequence of estimates that are guaranteed to converge to the desired integral. One of MCMC's great virtues is that it doesn't require too much effort: even novice users who need to calculate difficult integrals can quickly generate many MCMC algorithms that are guaranteed to work. Unfortunately, it is often the case that most of these algorithms are too inefficient to be useful. Thus, in practice users must spend time finding `good' MCMC algorithms. Adaptive MCMC (AMCMC) attempts to automate this process by iteratively refining the underlying MCMC algorithm as it improves its estimate of the integral, eventually learning both a good algorithm and a good estimate of the integral. When successful, AMCMC expands the range of problems for which MCMC methods can be applied without too much user effort.