Particle approximations of the score and observed information matrix in state space models with application to parameter estimation

Particle methods are popular computational tools for Bayesian inference in nonlinear non-Gaussian state space models. For this class of models, we present two particle algorithms to compute the score vector and observed information matrix recursively. The first algorithm is implemented with computational complexity <inline-formula><inline-graphic xlink:href="ASQ062IM1" xmlns:xlink="http://www.w3.org/1999/xlink"/></inline-formula> and the second with complexity <inline-formula><inline-graphic xlink:href="ASQ062IM2" xmlns:xlink="http://www.w3.org/1999/xlink"/></inline-formula>, where N is the number of particles. Although cheaper, the performance of the <inline-formula><inline-graphic xlink:href="ASQ062IM3" xmlns:xlink="http://www.w3.org/1999/xlink"/></inline-formula> method degrades quickly, as it relies on the approximation of a sequence of probability distributions whose dimension increases linearly with time. In particular, even under strong mixing assumptions, the variance of the estimates computed with the <inline-formula><inline-graphic xlink:href="ASQ062IM4" xmlns:xlink="http://www.w3.org/1999/xlink"/></inline-formula> method increases at least quadratically in time. The more expensive <inline-formula><inline-graphic xlink:href="ASQ062IM5" xmlns:xlink="http://www.w3.org/1999/xlink"/></inline-formula> method relies on a nonstandard particle implementation and does not suffer from this rapid degradation. It is shown how both methods can be used to perform batch and recursive parameter estimation. Copyright 2011, Oxford University Press.