Particle filters and Markov chains for learning of dynamical systems
Sammanfattning: Sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC) methods provide computational tools for systematic inference and learning in complex dynamical systems, such as nonlinear and non-Gaussian state-space models. This thesis builds upon several methodological advances within these classes of Monte Carlo methods.Particular emphasis is placed on the combination of SMC and MCMC in so called particle MCMC algorithms. These algorithms rely on SMC for generating samples from the often highly autocorrelated state-trajectory. A specific particle MCMC algorithm, referred to as particle Gibbs with ancestor sampling (PGAS), is suggested. By making use of backward sampling ideas, albeit implemented in a forward-only fashion, PGAS enjoys good mixing even when using seemingly few particles in the underlying SMC sampler. This results in a computationally competitive particle MCMC algorithm. As illustrated in this thesis, PGAS is a useful tool for both Bayesian and frequentistic parameter inference as well as for state smoothing. The PGAS sampler is successfully applied to the classical problem of Wiener system identification, and it is also used for inference in the challenging class of non-Markovian latent variable models.Many nonlinear models encountered in practice contain some tractable substructure. As a second problem considered in this thesis, we develop Monte Carlo methods capable of exploiting such substructures to obtain more accurate estimators than what is provided otherwise. For the filtering problem, this can be done by using the well known Rao-Blackwellized particle filter (RBPF). The RBPF is analysed in terms of asymptotic variance, resulting in an expression for the performance gain offered by Rao-Blackwellization. Furthermore, a Rao-Blackwellized particle smoother is derived, capable of addressing the smoothing problem in so called mixed linear/nonlinear state-space models. The idea of Rao-Blackwellization is also used to develop an online algorithm for Bayesian parameter inference in nonlinear state-space models with affine parameter dependencies.
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