  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.
