Jump-Markov state-space systems (JMSS) are widely used in statistical signal processing. However as is well known Bayesian restoration in JMSS is an NP-hard problem, so in practice all inference algorithms need to resort to some approximations. In this paper we focus on the computation of the conditional expectation of the hidden variable of interest given the available observations, which is optimal from the Bayesian quadratic risk viewpoint. We show that in some stochastic systems, namely the Partially Pairwise Markov-switching Chains (PPMSC) and Trees (PPMST), no approximation scheme is actually needed since the conditional expectation of interest (be it either in a filtering or prediction problem) can be computed exactly and in a number of operations linear in the number of observations. Index TermsBayesian restoration, Jump-Markov state-space systems, Partially Pairwise Markov switching models, NP-hard problems.

Abstract-Track-before-detect (TBD) aims at tracking trajectories of a target prior to detection by integrating raw measurements over time. Many TBD algorithms have been developed in the literature, based on the Hough Transform, Dynamic Programming or Maximum Likelihood estimation. However these methods fail in the case of maneuvering targets and/or non straight-line motion, or become very computationally expensive when the SNR gets low. Other techniques are based on the so-called switching or jump-Markov state-space system (JMSS) model. However, a drawback of JMSS is that it is not possible to perform exact Bayesian restoration. As a consequence, one has to resort to approximations such as particle filtering (PF). In this paper we propose an alternative method to approximate the optimal filter, which does not make use of Monte Carlo approximation. Our method is validated by computer simulations.