Weak convergence of adaptive Markov chain Monte Carlo

TL;DR

This paper establishes general conditions for the weak convergence of adaptive MCMC algorithms, enabling a broader estimation theory and extending convergence results to Wasserstein distances and unbounded functions.

Contribution

It introduces new weak convergence conditions for adaptive MCMC, extending previous total variation results and applying to diverse stochastic processes.

Findings

Weak convergence conditions for adaptive MCMC established

Weak law of large numbers proven for bounded and unbounded functions

Applications demonstrated in auto-regressive, Langevin, and Metropolis-Hastings processes

Abstract

This article develops general conditions for weak convergence of adaptive Markov chain Monte Carlo processes and is shown to imply a weak law of large numbers for bounded Lipschitz continuous functions. This allows an estimation theory for adaptive Markov chain Monte Carlo where previously developed theory in total variation may fail or be difficult to establish. Extensions of weak convergence to general Wasserstein distances are established along with a weak law of large numbers for possibly unbounded Lipschitz functions. Applications are applied to auto-regressive processes in various settings, unadjusted Langevin processes, and adaptive Metropolis-Hastings.