Backward Importance Sampling for Online Estimation of State Space Models

Abstract

This article proposes a new Sequential Monte Carlo algorithm to perform online estimation in the context of state space models when either the transition density of the latent state or the conditional likelihood of an observation given a state is intractable. In this setting, obtaining low variance estimators of expectations under the posterior distributions of the unobserved states given the observations is a challenging task. Following recent theoretical results for pseudo-marginal sequential Monte Carlo smoothers, a pseudo-marginal backward importance sampling step is introduced to estimate such expectations. This new step allows to reduce very significantly the computational time of the existing numerical solutions based on an acceptance–rejection procedure for similar performance, and to broaden the class of eligible models for such methods. For instance, in the context of multivariate stochastic differential equations, the proposed algorithm makes use of unbiased estimates of the unknown transition densities under much weaker assumptions than most standard alternatives. The performance of this estimator is assessed for high-dimensional discrete-time latent data models, for recursive maximum likelihood estimation in the context of Partially Observed Diffusion process (POD), and in the case of a bidimensional partially observed stochastic Lotka-Volterra model. Supplementary materials for this article are available online.

KEYWORDS:

• Hidden Markov models
• Online smoothing
• Sequential Monte Carlo

Supplementary Materials

Part of the code produced for the paper is available at: https://github.com/AMDonati/backward-IS.

Disclosure Statement

The authors report there are no competing interests to declare.

Notes

1 https://www.bgc-jena.mpg.de/wetter/