Uniform convergence over time of a nested particle filtering scheme for recursive parameter estimation in state–space Markov models

摘要

We analyse the performance of a recursive Monte Carlo method for the Bayesian estimation of the static parameters of a discrete–time state–space Markov model. The alg orithm employs two layers of particle filters to approximate the posterior probability distribution of the m odel parameters. In particular, the first layer yields an empirical distribution of samples on the paramete r space, while the filters in the second layer are auxiliary devices to approximate the (analytically intractab le) likelihood of the parameters. This approach relates the novel algorithm to the recent sequential Mo nte Carlo square (SMC 2 ) method, which provides a non-recursive solution to the same problem. In this paper, we investigate the appr oximation of integrals of real bounded functions with respect to the poster ior distribution of the system parameters. Under assumptions related to the compactness of the parameter support and the stability and continuity of the sequence of posterior distributions for the state–space m odel, we prove that the L p norms of the approximation errors vanish asymptotically (as the number of Mont e Carlo samples generated by the algorithm increases) and uniformly over time. We also prove that, un der the same assumptions, the proposed scheme can asymptotically identify the parameter values for a class of models. We conclude the paper with a numerical example that illustrates the uniform converg ence results by exploring the accuracy and stability of the proposed algorithm operating with long sequence s of observations.