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April 2015 Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms
Christophe Andrieu, Matti Vihola
Ann. Appl. Probab. 25(2): 1030-1077 (April 2015). DOI: 10.1214/14-AAP1022

Abstract

We study convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms (Andrieu and Roberts [Ann. Statist. 37 (2009) 697–725]). We find that the asymptotic variance of the pseudo-marginal algorithm is always at least as large as that of the marginal algorithm. We show that if the marginal chain admits a (right) spectral gap and the weights (normalised estimates of the target density) are uniformly bounded, then the pseudo-marginal chain has a spectral gap. In many cases, a similar result holds for the absolute spectral gap, which is equivalent to geometric ergodicity. We consider also unbounded weight distributions and recover polynomial convergence rates in more specific cases, when the marginal algorithm is uniformly ergodic or an independent Metropolis–Hastings or a random-walk Metropolis targeting a super-exponential density with regular contours. Our results on geometric and polynomial convergence rates imply central limit theorems. We also prove that under general conditions, the asymptotic variance of the pseudo-marginal algorithm converges to the asymptotic variance of the marginal algorithm if the accuracy of the estimators is increased.

Citation

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Christophe Andrieu. Matti Vihola. "Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms." Ann. Appl. Probab. 25 (2) 1030 - 1077, April 2015. https://doi.org/10.1214/14-AAP1022

Information

Published: April 2015
First available in Project Euclid: 19 February 2015

zbMATH: 1326.65012
MathSciNet: MR3313762
Digital Object Identifier: 10.1214/14-AAP1022

Subjects:
Primary: 65C40
Secondary: 60J05, 65C05

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.25 • No. 2 • April 2015
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