CONVERGENCE PROPERTIES OF PSEUDO-MARGINAL MARKOV CHAIN MONTE CARLO ALGORITHMS

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 mar...

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Bibliographic Details
Published in:The Annals of applied probability 2015-04, Vol.25 (2), p.1030-1077
Main Authors: Andrieu, Christophe, Vihola, Matti
Format: Article
Language:English
Subjects:
60J05
65C05
65C40
Algorithms
Asymptotic variance
Convergence
geometric ergodicity
Markov analysis
Markov chain Monte Carlo
Monte Carlo simulation
polynomial ergodicity
Probability distribution
pseudo-marginal algorithm
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Summary: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.
ISSN:1050-5164
2168-8737
DOI:10.1214/14-AAP1022