The Containment Condition and AdapFail algorithms

TL;DR

This paper analyzes adaptive MCMC algorithms, showing that failure to satisfy the Containment condition leads to poor performance, classifying such algorithms as AdapFail and advising against their use.

Contribution

It establishes a link between the Containment condition and algorithm efficiency, introducing the AdapFail class of poorly performing adaptive algorithms.

Findings

Failure to satisfy Containment causes poor convergence

AdapFail algorithms are less efficient than nonadaptive MCMC

Adaptive algorithms violating Containment should be avoided

Abstract

This short note investigates convergence of adaptive MCMC algorithms, i.e.\ algorithms which modify the Markov chain update probabilities on the fly. We focus on the Containment condition introduced in \cite{roberts2007coupling}. We show that if the Containment condition is \emph{not} satisfied, then the algorithm will perform very poorly. Specifically, with positive probability, the adaptive algorithm will be asymptotically less efficient then \emph{any} nonadaptive ergodic MCMC algorithm. We call such algorithms \texttt{AdapFail}, and conclude that they should not be used.