Large Sample Asymptotics of the Pseudo-Marginal Method

TL;DR

This paper investigates the asymptotic behavior of the pseudo-marginal algorithm as data size grows, providing guidelines for optimal scaling of proposals and Monte Carlo samples in large-sample settings.

Contribution

It proves the convergence of a rescaled pseudo-marginal chain to a limiting chain, validating and extending existing optimization results under large-sample conditions.

Findings

Convergence of the rescaled pseudo-marginal chain as data size increases.

Parameter-dependent guidelines for optimal proposal scaling.

Validation of existing optimization assumptions in large-sample regimes.

Abstract

The pseudo-marginal algorithm is a variant of the Metropolis--Hastings algorithm which samples asymptotically from a probability distribution when it is only possible to estimate unbiasedly an unnormalized version of its density. Practically, one has to trade-off the computational resources used to obtain this estimator against the asymptotic variances of the ergodic averages obtained by the pseudo-marginal algorithm. Recent works optimizing this trade-off rely on some strong assumptions which can cast doubts over their practical relevance. In particular, they all assume that the distribution of the difference between the log-density and its estimate is independent of the parameter value at which it is evaluated. Under regularity conditions we show here that, as the number of data points tends to infinity, a space-rescaled version of the pseudo-marginal chain converges weakly towards another pseudo-marginal chain for which this assumption indeed holds. A study of this limiting chain allows us to provide parameter dimension-dependent guidelines on how to optimally scale a normal random walk proposal and the number of Monte Carlo samples for the pseudo-marginal method in the large-sample regime. This complements and validates currently available results.