Characterization of uncertainty in Bayesian estimation using Sequential Monte Carlo methods

In estimation problems, accuracy of the estimates of the quantitiesof interest cannot be taken for granted. This means that estima-tion errors are expected, and a good estimation algorithm should beable not only to compute estimates that are optimal in some sense,but also provide meaningful measures of uncertainty associated withthose estimates. In some situations, we might also be able to reduceestimation uncertainty through the use of feedback on observations,an approach referred to as sensor management.Characterization of estimation uncertainty, as well as sensor manage-ment, are certainly difficult tasks for general partially observed pro-cesses, which might be non-linear, non-Gaussian, and/or have depen-dent process and observation noises. Sequential Monte Carlo (SMC)methods, also known as particle filters, are numerical Bayesian estima-tors which are, in principle, able to handle highly general estimationproblems. However, SMC methods are known to suffer from a phe-nomenon called degeneracy, or self-resolving, which greatly impairstheir usefulness against certain classes of problems.One of such classes, that we address in the first part of this the-sis, is the joint state and parameter estimation problem, where thereare unknown parameters to be estimated together with the time-varying state. Some SMC variants have been proposed to counter thedegeneracy phenomenon for this problem, but these state-of-the-arttechniques are either non-Bayesian or introduce biases on the systemmodel, which might not be appropriate if proper characterization ofestimation uncertainty is required. For this type of scenario, we pro-pose using the Rao-Blackwellized Marginal Particle Filter (RBMPF), a combination of two SMC algorithm variants: the Rao-BlackwellizedParticle Filter (RBPF) and the Marginal Particle Filter (MPF). Wederive two new versions of the RBMPF: one for models with low di-mensional parameter vectors, and another for more general models.We apply the proposed methods to two practical problems: the targettracking problem of turn rate estimation for a constant turn maneu-ver, and the econometrics problem of stochastic volatility estimation.Our proposed methods are shown to be effective solutions, both interms of estimation accuracy and statistical consistency, i.e. charac-terization of estimation uncertainty.Another problem where standard particle filters suffer from degener-acy, addressed in the second part of this thesis, is the joint multi-targettracking and labelling problem. In comparison with the joint state andparameter estimation problem, this problem poses an additional chal-lenge, namely, the fact that it has not been properly mathematicallyformulated in previous literature. Using Finite Set Statistics (FISST),we provide a sound theoretical formulation for the problem, and inorder to actually solve the problem, we propose a novel Bayesian al-gorithm, the Labelling Uncertainty-Aware Particle Filter (LUA-PF)filter, essentially a combination of the RBMPF and the Multi-targetSequential Monte Carlo (M-SMC) filter techniques. We show that thenew algorithm achieves significant improvements on both finding thecorrect track labelling and providing a meaningful measure of labellinguncertainty.In the last part of this thesis, we address the sensor managementproblem. Although we apply particle filters to the problem, they arenot the main focus of this part of the work. Instead, we concentrateon a more fundamental question, namely, which sensor managementcriterion should be used in order to obtain the best results in terms ofinformation gain and/or reduction of uncertainty. In order to answerthis question, we perform an in-depth theoretical and empirical anal-ysis on two popular sensor management criteria based on information theory – the Kullback-Leibler and Rényi divergences. On the basisof this analysis, we are able to either confirm or reject some previousarguments used as theoretical justification for these two criteria.