flavour of the MALA and the mMALA, resulting in the two methods proposed in this paper, PMH1 and PMH2, respectively. In particular, we make use of a fixed-lag (FL) particle smoother [@KitagawaSato2001] to estimate the gradient and Hessian. The motivation for this is that this smoother only makes use of the weighted particles computed by the particle filter. Consequently, we obtain this information as a *byproduct* of the likelihood computation in the PMH algorithm. This results in only a small computational overhead for the proposed methods when compared to the marginal method.

Finally, we provide numerical experiments to illustrate the benefits of using the gradient and Hessian and the accuracy of the FL smoother. We demonstrate some interesting properties of the proposed algorithms, in particular that they enjoy (i) a shorter burn-in compared with the marginal algorithm, (ii) a better mixing of the Markov chain in the stationary phase, and (iii) a simplified tuning of the step length(s), especially when the target distribution is non-isotropic.

Particle Metropolis-Hastings {#sec:overview}
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In this section, we review the PMH algorithm and show how the random variables used to compute the likelihood estimator can be incorporated in the proposal construction. We also outline the 