Particle-based methods for parameter estimation and tracking: Numerical experiments

The purpose of this work is to obtain as much intuition as possible, through numerical experiments in a simple case where exact solutions are explicitly available, about the particle approximation of finite signed measures. A prototypical example of a finite signed measure is the derivative, w.r.t. a parameter of the model, of some probability distributions related with a hidden Markov chain. This includes prior, prediction, filtering probability distributions, etc. Two points of view are considered here, to feel the quality of the approximation, at least in a qualitative manner: (i) how accurate is the particle approximation of the finite signed measure, in view of an histogram representation of the weighted particle system? and (ii) considering the log-likelihood function and the score function, how close is the approximate expression provided by the particle approximation to the exact expression? These two questions seem closely related, however the numerical experiments presented in this work show that one of the two particle approximation schemes fails to satisfy the first criteria (quality of the approximation of the finite signed measure), and that both schemes satisfy the second criteria (quality of the approximation of the statistics).