🤖 AI Summary
This work addresses the challenge of verifying ergodicity for adaptive MCMC algorithms in non-compact state and parameter spaces, where traditional compactness assumptions fail. By abandoning such restrictive assumptions, the authors instead introduce probabilistic bounds on the sample and parameter sequences to formulate a new set of easily verifiable sufficient conditions. Integrating tools from MCMC theory, adaptive algorithm analysis, and concentration inequalities, they establish a novel ergodicity framework that operates without compactness requirements. This approach significantly enhances the practical applicability and tractability of convergence analysis in complex real-world settings where non-compactness is inherent.
📝 Abstract
This paper provides sufficient conditions over the sequence of samples and parameters of an adaptive Markov Chain Monte Carlo (MCMC) algorithm to converge to the target distribution. These conditions aim to make more easily usable classical conditions formulated over the transition kernels, without needing, as was done in other works, to assume the compactness of both sample and parameter spaces. The condition of compactness is replaced here with a probability bound over the sequence of both samples and parameters.