On the ergodicity of the adaptive Metropolis algorithm on unbounded domains

TL;DR

This paper establishes conditions under which the adaptive Metropolis algorithm remains ergodic on unbounded domains, focusing on target distributions with super-exponentially decaying tails and regular contours.

Contribution

It provides new sufficient conditions ensuring the ergodicity of the adaptive Metropolis algorithm for noncompact support distributions, extending previous results.

Findings

Ergodicity holds if target tails decay super-exponentially.

Regular contours of the target distribution are required.

The approach involves auxiliary constrained processes and geometric drift analysis.

Abstract

This paper describes sufficient conditions to ensure the correct ergodicity of the Adaptive Metropolis (AM) algorithm of Haario, Saksman and Tamminen [Bernoulli 7 (2001) 223--242] for target distributions with a noncompact support. The conditions ensuring a strong law of large numbers require that the tails of the target density decay super-exponentially and have regular contours. The result is based on the ergodicity of an auxiliary process that is sequentially constrained to feasible adaptation sets, independent estimates of the growth rate of the AM chain and the corresponding geometric drift constants. The ergodicity result of the constrained process is obtained through a modification of the approach due to Andrieu and Moulines [Ann. Appl. Probab. 16 (2006) 1462--1505].