  Pseudo-marginal Metropolis-Hastings (pmMH) is a powerful method for Bayesian
inference in models where the posterior distribution is analytical intractable
or computationally costly to evaluate directly. It operates by introducing
additional auxiliary variables into the model and form an extended target
distribution, which then can be evaluated point-wise. In many cases, the
standard Metropolis-Hastings is then applied to sample from the extended target
and the sought posterior can be obtained by marginalisation. However, in some
implementations this approach suffers from poor mixing as the auxiliary
variables are sampled from an independent proposal. We propose a modification
to the pmMH algorithm in which a Crank-Nicolson (CN) proposal is used instead.
This results in that we introduce a positive correlation in the auxiliary
variables. We investigate how to tune the CN proposal and its impact on the
mixing of the resulting pmMH sampler. The conclusion is that the proposed
modification can have a beneficial effect on both the mixing of the Markov
chain and the computational cost for each iteration of the pmMH algorithm.
