On the use of sequential Monte Carlo methods for approximating smoothing functionals, with application to fixed parameter estimation

¹ Center for Mathematical Sciences, Lund University, Sweden
² Laboratoire Traitement et Communication de l'Information, CNRS / GET Télécom Paris, France
³ Centre de Mathématiques APpliquées, Ecole Polytechnique, France

Abstract

Sequential Monte Carlo (SMC) methods have demonstrated a strong potential for inference on the state variables in Bayesian dynamic models. In this context, it is also often needed to calibrate model parameters. To do so, we consider block maximum likelihood estimation based either on EM (Expectation-Maximization) or gradient methods. In this approach, the key ingredient is the computation of smoothed sum functionals of the hidden states, for a given value of the model parameters. It has been observed by several authors that using standard SMC methods for this smoothing task requires a substantial number of particles and may be unreliable for larger observation sample sizes. We introduce a simple variant of the basic sequential smoothing approach based on forgetting ideas. This modification, which is transparent in terms of computation time, reduces the variability of the approximation of the sum functional. Under suitable regularity assumptions, it is shown that this modification indeed allows a tighter control of the L_p error of the approximation.