The Containment Condition and AdapFail algorithms

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.