We present a novel sequential Monte Carlo approach to online smoothing of additive functionals in a very general class of path-space models. Hitherto, the solutions proposed in the literature suffer from either long-term numerical instability due to particle-path degeneracy or, in the case that degeneracy is remedied by particle approximation of the so-called backward kernel, high computational demands. In order to balance optimally computational speed against numerical stability, we propose to furnish a (fast) naive particle smoother, propagating recursively a sample of particles and associated smoothing statistics, with an adaptive backward-sampling-based updating rule which allows the number of (costly) backward samples to be kept at a minimum. This yields a new, function-specific additive smoothing algorithm, AdaSmooth, which is computationally fast, numerically stable and easy to implement. The algorithm is provided with rigorous theoretical results guaranteeing its consistency, asymptotic normality and long-term stability as well as numerical results demonstrating empirically the clear superiority of AdaSmooth to existing algorithms. for this article are available online.

This paper presents a novel algorithm for efficient online estimation of the filter derivatives in general hidden Markov models. The algorithm, which has a linear computational complexity and very limited memory requirements, is furnished with a number of convergence results, including a central limit theorem with an asymptotic variance that can be shown to be uniformly bounded in time. Using the proposed filter derivative estimator, we design a recursive maximum likelihood algorithm updating the parameters according the gradient of the one-step predictor log-likelihood. The efficiency of this online parameter estimation scheme is illustrated in a simulation study.

We present a novel algorithm, an adaptive-lag smoother, approximating efficiently, in an online fashion, sequences of expectations under the marginal smoothing distributions in general state-space models. The algorithm evolves recursively a bank of estimators, one for each marginal, in resemblance with the so-called particle-based, rapid incremental smoother (PaRIS). Each estimator is propagated until a stopping criterion, measuring the fluctuations of the estimates, is met. The presented algorithm is furnished with theoretical results describing its asymptotic limit and memory usage.

This paper presents a novel algorithm, the particle-based, rapid incremental smoother (PaRIS), for efficient online approximation of smoothed expectations of additive state functionals in general hidden Markov models. The algorithm, which has a linear computational complexity under weak assumptions and very limited memory requirements, is furnished with a number of convergence results, including a central limit theorem. An interesting feature of PaRIS, which samples on-the-fly from the retrospective dynamics induced by the particle filter, is that it requires two or more backward draws per particle in order to cope with degeneracy of the sampled trajectories and to stay numerically stable in the long run with an asymptotic variance that grows only linearly with time.

Estimating online the parameters of general state-space hidden Markov models is a topic of importance in many scientific and engineering disciplines. In this paper we present an online parameter estimation algorithm obtained by casting our recently proposed particle-based, rapid incremental smoother (Paris) into the framework of recursive maximum likelihood estimation for general hidden Markov models. Previous such particle implementations suffer from either quadratic complexity in the number of particles or from the well-known degeneracy of the genealogical particle paths. By using the computational efficient and numerically stable Paris algorithm for estimating the needed prediction filter derivatives we obtain a fast algorithm with a computational complexity that grows only linearly with the number of particles. The efficiency and stability of the proposed algorithm are illustrated in a simulation study.

Estimating the parameters of general state-space models is a topic of importance for many scientific and engineering disciplines. In this paper we present an online parameter estimation algorithm obtained by casting our recently proposed particle-based, rapid incremental smoother (PaRIS) into the framework of online expectation-maximization (EM) for state-space models proposed by Cappé (2011). Previous such particle-based implementations of online EM suffer typically from either the well-known degeneracy of the genealogical particle paths or a quadratic complexity in the number of particles. However, by using the computationally efficient and numerically stable PaRIS algorithm for estimating smoothed expectations of timeaveraged sufficient statistics of the model we obtain a fast algorithm with very limited memory requirements and a computational complexity that grows only linearly with the number of particles. The efficiency of the algorithm is illustrated in a simulation study.

This paper deals with the problem of estimating expectations of sums of additive functionals under the joint smoothing distribution in general hidden Markov models. Computing such expectations is a key ingredient in any kind of expectation-maximization-based parameter inference in models of this sort. The paper presents a computationally efficient algorithm for online estimation of these expectations in a forward manner. The proposed algorithm has a linear computational complexity in the number of particles and does not require old particles and weights to be stored during the computations. The algorithm avoids completely the well-known particle path degeneracy problem of the standard forward smoother. This makes it highly applicable within the framework of online expectation-maximization methods. The simulations show that the proposed algorithm provides the same precision as existing algorithms at a considerably lower computational cost.