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The Containment Condition and Adapfail Algorithms

Part of: Markov processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  30 January 2018

Krzysztof Łatuszyński  and
Jeffrey S. Rosenthal
Krzysztof Łatuszyński*
Affiliation:
University of Warwick
Jeffrey S. Rosenthal*
Affiliation:
University of Toronto
*
∗ Postal address: Department of Statistics, University of Warwick, Coventry CV4 7AL, UK. Email address: latuch@gmail.com
∗∗ Postal address: Department of Statistics, University of Toronto, Toronto, Ontario M5S 3G3, Canada. Email address: jeff@math.toronto.edu
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Abstract

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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 Roberts and Rosenthal (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.

Keywords

Markov chain Monte Carloadaptive MCMCcontainment conditionergodicityconvergence rate

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 65C05: Monte Carlo methods
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 4 , December 2014 , pp. 1189 - 1195
Copyright
© Applied Probability Trust 

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