This thesis presents methods for the inference of system state and the learning of model structure for a number of hidden-state time series models, within a Bayesian probabilistic framework. Motivating examples are taken from application areas including finance, physical object tracking and audio restoration. The work in this thesis can be broadly divided into three themes: system and parameter estimation in linear jump-diffusion systems, non-parametric model (system) estimation and batch audio restoration.

For linear jump-diffusion systems, efficient state estimation methods based on the variable rate particle filter are presented for the general linear case (chapter 3) and a new method of parameter estimation based on Particle MCMC methods is introduced and tested against an alternative method using reversible-jump MCMC (chapter 4).

Non-parametric model estimation is examined in two settings: the estimation of non-parametric environment models in a SLAM-style problem, and the estimation of the network structure and forms of linkage between multiple objects. In the former case, a non-parametric Gaussian process prior model is used to learn a potential field model of the environment in which a target moves. Efficient solution methods based on Rao-Blackwellized particle filters are given (chapter 5). In the latter case, a new way of learning non-linear inter-object relationships in multi-object systems is developed, allowing complicated inter-object dynamics to be learnt and causality between objects to be inferred. Again based on Gaussian process prior assumptions, the method allows the identification of a wide range of relationships between objects with minimal assumptions and admits efficient solution, albeit in batch form at present (chapter 6).

Finally, the thesis presents some new results in the restoration of audio signals, in particular the removal of impulse noise (pops and clicks) from audio recordings (chapter 7)