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

Abstract

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 O(N ) and the second with complexity O(N 2 ), where N is the number of particles. Although cheaper, the performance of the O(N ) 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 O(N ) method increases at least quadratically in time. The more expensive O(N 2 ) 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.

A backward particle interpretation of Feynman-Kac formulae

“…(Admittedly, this growth was established in Section 8 under stringent mixing assumptions.) This has indeed been demonstrated in similar estimation problems involving models for which mixing has not been established [26].…”

“…For instance, it may be used to implement an on-line version of the Expectation-Maximization algorithm as detailed in [23], Section 3.2. In a different approach to recursive parameter estimation, an on-line particle algorithm is presented in [26] to compute the score for non-linear non-Gaussian state-space models. In fact, the algorithm of [26] is actually implementing a special case of the above recursion and may be reinterpreted as an "on-the-fly" computation of the forward filtering backward smoothing estimate of an additive functional derived from Fisher's identity.…”

“…In a different approach to recursive parameter estimation, an on-line particle algorithm is presented in [26] to compute the score for non-linear non-Gaussian state-space models. In fact, the algorithm of [26] is actually implementing a special case of the above recursion and may be reinterpreted as an "on-the-fly" computation of the forward filtering backward smoothing estimate of an additive functional derived from Fisher's identity. The convergence analysis of the N -particle measures Q N n towards their limiting value Q n , as N → ∞, is intimately related to the convergence of the flow of particle measures (η N p ) 0≤p≤n towards their limiting measures (η p ) 0≤p≤n .…”

“…Additionally, we show how the forward filtering backward smoothing estimates of additive functionals can be computed using a forward only recursion. This has applications to online parameter estimation for non-linear non-Gaussian state-space models [13,26].…”

Particle metropolis hastings using Langevin dynamics

“…Methods for score estimation using particle filters have previous been proposed and used for maximum likelihood inference in e.g. [9,10,11,12,13].…”