Particle Metropolis-Hastings using gradient and Hessian information

Particle Metropolis-Hastings (PMH) allows for Bayesian parameter in-ference in nonlinear state space models by combining MCMC and particle filtering. The latter is used to estimate the intractable likelihood. In its original formulation, PMH makes use of a marginal MCMC proposal for the parameters, typically a Gaussian random walk. However, this can lead to a poor exploration of the parameter space and an inefficient use of the generated particles. We propose two alternative versions of PMH that incorporate gradi-ent and Hessian information about the posterior into the proposal. This information is more or less obtained as a byproduct of the likelihood esti-mation. Indeed, we show how to estimate the required information using a fixed-lag particle smoother, with a computational cost growing linearly in the number of particles. We conclude that the proposed methods can: (i) decrease the length of the burn-in phase, (ii) increase the mixing of the Markov chain at the stationary phase, and (iii) make the proposal distribution scale invariant which simplifies tuning.