Unbiased approximations of products of expectations

Anthony Lee, Simone Tiberi, Giacomo Zanella

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

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We consider the problem of approximating the product of n expectations with respect to a common probability distribution μ. Such products routinely arise in statistics as values of the likelihood in latent variable models. Motivated by pseudo-marginal Markov chain Monte Carlo schemes, we focus on unbiased estimators of such products. The standard approach is to sample N particles from μ and assign each particle to one of the expectations. This is wasteful and typically requires the number of particles to grow quadratically with the number of expectations. We propose an alternative estimator that approximates each expectation using most of the particles while preserving unbiasedness. We carefully study its properties, showing that in latent variable contexts the proposed estimator needs only O(n) particles to match the performance of the standard approach with O(n2) particles. We demonstrate the procedure on two latent variable examples from approximate Bayesian computation and single-cell gene expression analysis, observing computational gains of the order of the number of expectations, i.e. data points, n.
Original languageEnglish
Article numberasz008
Number of pages8
Issue number3
Early online date8 Apr 2019
Publication statusPublished - Sep 2019


  • latent variables
  • Markov chain Monte Carlo
  • pseudo-marginal
  • approximate Bayesian computation

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