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

Correlated pseudo-marginal Metropolis-Hastings using quasi-Newton proposals