Advances in Sequential Monte Carlo-based Statistical Learning: Online Algorithmic and Variational Inference

QC 2024-11-14

Abstract

This dissertation deals with sequential Monte Carlo (SMC) methods, also known as particle filters, focusing on online statistical learning in general state-space models (SSMs). SSMs are today an invaluable tool for modelling serial data in a variety of scientific and engineering disciplines such as automatic control, signal processing, biology, and finance. By including latent, Markovian states, SSMs offer high modeling flexibility, while SMC methods, which propagate recursively weighted samples using importance sampling and resampling techniques, are well-suited for state and model-parameter inference. This thesis, which consists of four papers, contributes to the enhancement of the online functionality of SMC, where 'online' refers to numerically stable estimation procedures with constant computational complexity and constant memory requirements over time.

The contribution of the thesis can be divided into two parts. The first part, which spans two papers, deals with algorithmic inference, focusing on both Bayesian state inference via SMC algorithms and the evaluation of their accuracy. Both papers address the challenges caused by the phenomenon of so-called particle-path degeneracy, an unavoidable issue caused by the resampling operation in SMC algorithms. Paper A presents AdaSmooth, a new algorithm developed to approximate expectations under joint-state posteriors in general SSMs when the objective is additive in the states. AdaSmooth, which uses adaptive backward-sampling techniques, avoids path-degeneracy and provides numerically stable estimates over infinite time horizons with reduced computational demands. Paper B presents ALVaR, an online method that consistently estimates the asymptotic variance of a particle filter based solely on the particle genealogy, avoiding extra sampling.

The second part combines SMC methods with variational inference and aims to develop online algorithms for parameter estimation in SSMs and proposal adaptation in particle filters, the latter being essential for accurate state posterior estimates. Paper C introduces OVSMC, which extends so-called variational SMC to online settings and allows simultaneous estimation of unknown model parameters and proposal kernel optimisation. Paper D proposes OSIWAE, which performs online variational inference by optimising a lower bound on the time-normalised limiting log-likelihood, resulting in a more theoretically grounded approach than OVSMC. OSIWAE ideally requires access to the filter state posteriors and their derivatives, which lack closed-form expressions in general. For this reason, a particle-based version, SMC-OSIWAE, which estimates the filter derivatives using AdaSmooth from Paper A, is developed.

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