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K Latuszynski and JS Rosenthal

The Containment Condition and AdapFail algorithms

Abstract: This short note investigates convergence of adaptive MCMCalgorithms, i.e. algorithms which modify the Markov chain update probabilities on the y. We focus on the Containment condition introduced in[RR07]. We show that if the Containment condition is not satisfied, thenthe algorithm will perform very poorly. Specifically, with positive probability, the adaptive algorithm will be asymptotically less efficient then anynonadaptive ergodic MCMC algorithm. We call such algorithms AdapFail,and conclude that they should not be used. AMS 2000 subject classifications: Primary 60J05, 65C05.

Keywords: Markov chain Monte Carlo, adaptive MCMC, Containment condition, ergodicity, convergence rates.