Title: Large Sample Asymptotics of the Pseudo-Marginal Method,
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 additive error in the log-likelihood estimator is independent of the
parameter value at which it is evaluated. Under weak 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.