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Efficiency of reversible MCMC methods: elementary derivations and applications to composite methods

Part of: Markov processes

Published online by Cambridge University Press:  18 September 2024

Radford M. Neal  and
Jeffrey S. Rosenthal Open the ORCID record for Jeffrey S. Rosenthal [Opens in a new window]
Radford M. Neal*
Affiliation:
University of Toronto
Jeffrey S. Rosenthal*
Affiliation:
University of Toronto
*
*Postal address: Department of Statistical Sciences, University of Toronto.
*Postal address: Department of Statistical Sciences, University of Toronto.
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Abstract

We review criteria for comparing the efficiency of Markov chain Monte Carlo (MCMC) methods with respect to the asymptotic variance of estimates of expectations of functions of state, and show how such criteria can justify ways of combining improvements to MCMC methods. We say that a chain on a finite state space with transition matrix P efficiency-dominates one with transition matrix Q if for every function of state it has lower (or equal) asymptotic variance. We give elementary proofs of some previous results regarding efficiency dominance, leading to a self-contained demonstration that a reversible chain with transition matrix P efficiency-dominates a reversible chain with transition matrix Q if and only if none of the eigenvalues of $Q-P$ are negative. This allows us to conclude that modifying a reversible MCMC method to improve its efficiency will also improve the efficiency of a method that randomly chooses either this or some other reversible method, and to conclude that improving the efficiency of a reversible update for one component of state (as in Gibbs sampling) will improve the overall efficiency of a reversible method that combines this and other updates. It also explains how antithetic MCMC can be more efficient than independent and identically distributed sampling. We also establish conditions that can guarantee that a method is not efficiency-dominated by any other method.

Keywords

Markov chainasymptotic varianceefficiency dominationPeskun ordering

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J22: Computational methods in Markov chains

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 1 , March 2025 , pp. 188 - 208
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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