On the stability and ergodicity of adaptive scaling Metropolis algorithms

TL;DR

This paper analyzes the stability and ergodicity of two adaptive Metropolis algorithms that adjust proposal scaling and covariance without predefined bounds, proving strong law of large numbers under certain conditions.

Contribution

It introduces two adaptive Metropolis algorithms with unbounded scaling parameters and proves their stability and ergodicity under broad conditions.

Findings

Both algorithms are proven to be stable and ergodic.

A strong law of large numbers is established for the algorithms.

The algorithms work with target densities having compact support or super-exponentially decaying tails.

Abstract

The stability and ergodicity properties of two adaptive random walk Metropolis algorithms are considered. The both algorithms adjust the scaling of the proposal distribution continuously based on the observed acceptance probability. Unlike the previously proposed forms of the algorithms, the adapted scaling parameter is not constrained within a predefined compact interval. The first algorithm is based on scale adaptation only, while the second one incorporates also covariance adaptation. A strong law of large numbers is shown to hold assuming that the target density is smooth enough and has either compact support or super-exponentially decaying tails.