Long-term stability of sequential Monte Carlo methods under verifiable conditions

Randal Douc (1, 2, 3) , Éric Moulines (4) , Jimmy Olsson (5)

1 TSP - CITI - Communications, Images et Traitement de l'Information
2 SAMOVAR - Services répartis, Architectures, MOdélisation, Validation, Administration des Réseaux
3 CNRS - Centre National de la Recherche Scientifique
4 LTCI - Laboratoire Traitement et Communication de l'Information
5 Skane University Hospital [Lund]

This paper discusses particle filtering in general hidden Markov models (HMMs) and presents novel theoretical results on the longterm stability of bootstrap-type particle filters. More specifically, we establish that the asymptotic variance of the Monte Carlo estimates produced by the bootstrap filter is uniformly bounded in time. On the contrary to most previous results of this type, which in general presuppose that the state space of the hidden state process is compact (an assumption that is rarely satisfied in practice), our very mild assumptions are satisfied for a large class of HMMs with possibly noncompact state space. In addition, we derive a similar time uniform bound on the asymptotic Lp error. Importantly, our results hold for misspecified models; that is, we do not at all assume that the data entering into the particle filter originate from the model governing the dynamics of the particles or not even from an HMM

Mots clés: Asymptotic variance; General hidden Markov models; Local Doeblin condition; Bootstrap particle filter; Sequential Monte Carlo methods; Time uniform convergence

Domaines: Probabilités [math.PR]

Randal Douc, Éric Moulines, Jimmy Olsson. Long-term stability of sequential Monte Carlo methods under verifiable conditions. The Annals of Applied Probability, 2014, 24 (5), pp.1767 - 1802. ⟨10.1214/13-AAP962⟩. ⟨hal-01262408⟩