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

Documento depositado en el repositorio arXiv.org. Versión: arXiv:1603.09005v1 [stat.CO]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 algorithm employs two layers of particle filters to approximate the posterior probability distribution of the model parameters. In particular, the first layer yields an empirical distribution of samples on the parameter space, while the filters in the second layer are auxiliary devices to approximate the (analytically intractable) likelihood of the parameters. This approach relates the this algorithm to the recent sequential Monte Carlo square (SMC2) method, which provides a {\em non-recursive} solution to the same problem. In this paper, we investigate the approximation, via the proposed scheme, of integrals of real bounded functions with respect to the posterior 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 model, we prove that the Lp norms of the approximation errors vanish asymptotically (as the number of Monte Carlo samples generated by the algorithm increases) and uniformly over time. We also prove that, under 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 convergence results by exploring the accuracy and stability of the proposed algorithm operating with long sequences of observations.The work of J. Míguez was partially supported by Ministerio de Economía y Competitividad of Spain (project TEC2012-38883-C02-01 COMPREHENSION) and the Office of Naval Research Global (award no. N62909-15-1-2011). Part of this work was carried out while J. M. was a visitor at the Department of Mathematics of Imperial College London, with partial support from an EPSRC Mathematics Platform grant. D. C. and J. M. would also like to acknowledge the support of the Isaac Newton Institute through the program “Monte Carlo Inference for High-Dimensional Statistical Models”