As two fundamental pillars of Bayesian inference for time series, stochastic modelling and approximate Bayesian inference play crucial roles in providing accurate priors for underlying random processes and addressing the challenges of evaluating posterior distributions when exact computation is infeasible. Balancing novel contributions in both areas, this thesis highlights innovative stochastic models in Chapters 2 and 3, and puts emphasis on novel approximate Bayesian inference schemes in Chapters 4, 5, and 6. Driven by applications in object tracking and intent inference, the developed methodologies aim to accurately capture desired motion characteristics while enhancing the effectiveness, efficiency, and robustness of estimations.

The applications of intent inference and single object tracking are considered in Chapters 2 and 3. Chapter 2 presents a generic Bayesian intent inference framework capable of predicting the destination of a tracked object, along with an exploration of several mean-reverting stochastic processes that serve as dynamic models within the framework. Chapter 3 develops novel α-stable Lévy state-space models for manoeuvring object tracking and intent prediction, expressed in continuous time as Lévy processes. These models effectively capture sharp changes in state induced by erratic maneuvers with heavy-tailed α-stable driven noise, while maintaining an advantageous conditionally Gaussian transition. Additionally, this chapter introduces an efficient intent inference procedure that accommodates dynamically varying intent across the surveyed area, offering versatile solutions for diverse tracking scenarios.

Chapter 4 introduces a novel conditionally factorised variational family that retains dependence between desired variables at user-defined levels of detail. A new variational Bayes algorithm is then proposed and implemented with importance sampling. It guarantees a better variational lower bound by choosing a finer conditional structure, offering a flexible trade-off between computational cost and inference accuracy.

Multi-object tracking tasks are addressed in Chapters 5 and 6 with Poisson measurement processes. Chapter 5 introduces a variational Bayes multi-object tracker that effectively performs tracking, data association, and learning of target and clutter rates, while offering substantial efficiency gains and parallelisable implementation. Chapter 6 extends this tracker to tackle highly challenging tracking scenarios involving a large number of closely-spaced objects and heavy clutter. By introducing a novel variational localisation strategy that quickly rediscovers missed targets under extremely heavy clutter, the enhanced tracker can automatically detect and recover from track loss, delivering outstanding performance in tracking accuracy and efficiency under difficult tracking conditions.