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.
Categories: stat.CO