On the Convergence Rates of Some Adaptive Markov Chain Monte Carlo Algorithms

This paper studies the mixing time of certain adaptive Markov Chain Monte Carlo algorithms. Under some regularity conditions, we show that the convergence rate of Importance Resampling MCMC (IRMCMC) algorithm, measured in terms of the total variation distance is $O(n^{-1})$, and by means of an example, we establish that in general, this algorithm does not converge at a faster rate. We also study the Equi-Energy sampler and establish that its mixing time is of order $O(n^{-1/2})$.

💡 Research Summary

This paper provides a rigorous theoretical investigation of the mixing times of two adaptive Markov chain Monte Carlo (MCMC) algorithms: Importance Resampling MCMC (IRMCMC) and the Equi‑Energy (EE) sampler. The authors adopt total variation distance as the metric for convergence and work under a set of standard regularity conditions that are common in the adaptive MCMC literature, namely diminishing adaptation and containment.

The first part of the paper introduces the adaptive MCMC framework. Unlike classical MCMC, where the transition kernel is fixed, adaptive schemes update the kernel on‑the‑fly using information gathered from the chain’s past trajectory. The two conditions mentioned above guarantee that the adaptation does not destroy the Markov property in the limit and that the chain remains in a “well‑behaved” region of the state space with high probability.

For IRMCMC, the algorithm proceeds by generating a set of candidate points at each iteration, assigning each a weight proportional to the target density (or an importance function), and then resampling from this weighted set. This resampling step can be viewed as a stochastic approximation to the target distribution. The authors model the sequence of transition kernels ({P_n}) induced by the adaptive procedure and prove that, under the assumption that the target density is bounded away from zero and infinity on a compact support and that the importance weights are uniformly bounded, the total variation distance satisfies
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