Introduction to "Particle Metropolis-Hastings using gradient and Hessian information" by J. Dahlin, F. Lindsten, T. Schön

The authors investigate the use of metropolis adjusted Langevin algorithms (MALA) in the context of particle MCMC algorithms. The ability to use this type of updates can lead to more efficient MCMC algorithms. The challenge in this context is that MALA and more sophisticated versions require the evaluation of the gradient of the loglikelihood and/or its Hessian, which are not available analytically. A way around this consists of estimating these quantities numerically. This can be achieved efficiently by exploiting the particle filter output used to compute the estimator of the likelihood function required to implement particle MCMCs. This idea was proposed originally in Doucet et al. (2011) and similar ideas have been subsequently investigated in Nemeth and Fearnhead (2014) and Dahlin et al. (2013) and Dahlin et al. (2014). The authors’ contribution here is to propose solutions to the fact that the estimated Hessian can be negative and an extensive numerical evaluation of the performance properties of the algorithmwhich results in some guidelines concerning