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The containment condition and AdapFail algorithms. (English) Zbl 1329.60263
Summary: This short note investigates convergence of adaptive Markov chain Monte Carlo algorithms, i.e. algorithms which modify the Markov chain update probabilities on the fly. We focus on the containment condition introduced by Roberts and Rosenthal in 2007. We show that if the containment condition is not satisfied, then the algorithm will perform very poorly. Specifically, with positive probability, the adaptive algorithm will be asymptotically less efficient then any nonadaptive ergodic MCMC algorithm. We call such algorithms AdapFail, and conclude that they should not be used.

60J22 Computational methods in Markov chains
60J05 Discrete-time Markov processes on general state spaces
65C05 Monte Carlo methods
65C40 Numerical analysis or methods applied to Markov chains
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