Air Markov Chain Monte Carlo

TL;DR

This paper introduces AirMCMC, a class of adaptive algorithms where the Markov kernel is periodically changed based on the entire chain output, simplifying analysis and maintaining performance.

Contribution

It proposes AirMCMC algorithms with changing kernels at increasing intervals and proves their convergence properties under various drift conditions.

Findings

Proves $L_2$-convergence, WLLN, SLLN, and CLT for AirMCMC.

Shows many adaptive MCMC algorithms can be transformed into Air versions.

Empirical evidence indicates performance remains comparable.

Abstract

We introduce a class of Adapted Increasingly Rarely Markov Chain Monte Carlo (AirMCMC) algorithms where the underlying Markov kernel is allowed to be changed based on the whole available chain output but only at specific time points separated by an increasing number of iterations. The main motivation is the ease of analysis of such algorithms. Under the assumption of either simultaneous or (weaker) local simultaneous geometric drift condition, or simultaneous polynomial drift we prove the $L_2-$convergence, Weak and Strong Laws of Large Numbers (WLLN, SLLN), Central Limit Theorem (CLT), and discuss how our approach extends the existing results. We argue that many of the known Adaptive MCMC algorithms may be transformed into the corresponding Air versions, and provide an empirical evidence that performance of the Air version stays virtually the same.