  Pseudo-marginal Metropolis-Hastings (pmMH) is a versatile algorithm for
sampling from target distributions which are not easy to evaluate point-wise.
However, pmMH requires good proposal distributions to sample efficiently from
the target, which can be problematic to construct in practice. This is
especially a problem for high-dimensional targets when the standard random-walk
proposal is inefficient. We extend pmMH to allow for constructing the proposal
based on information from multiple past iterations. As a consequence,
quasi-Newton (qN) methods can be employed to form proposals which utilize
gradient information to guide the Markov chain to areas of high probability and
to construct approximations of the local curvature to scale step sizes. The
proposed method is demonstrated on several problems which indicate that qN
proposals can perform better than other common Hessian-based proposals.
