Convergence properties of pseudo-marginal markov chain monte carlo algorithms

Christophe Andrieu, Matti Vihola

Research output: Contribution to journalArticle (Academic Journal)peer-review

40 Citations (Scopus)
316 Downloads (Pure)


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 pseudomarginal algorithm converges to the asymptotic variance of the marginal algorithm if the accuracy of the estimators is increased.

Original languageEnglish
Pages (from-to)1030-1077
Number of pages48
JournalAnnals of Applied Probability
Issue number2
Publication statusPublished - 1 Apr 2015


  • Asymptotic variance
  • Geometric ergodicity
  • Markov chain Monte Carlo
  • Polynomial ergodicity
  • Pseudo-marginal algorithm


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