  This paper 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 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, and in the case of a bidimensional partially observed stochastic
Lotka-Volterra model.
