Sequential Bayesian Learning for State Space Models

Abstract

This thesis explores the topic of sequential inference on a variety of novel model classes. Chapter 2 focuses on a class of discrete State Space Models (SSM) known as Hidden Semi-Markov Model (HSMM), a versatile generalization of the famous Hidden Markov Model (HMM) in which the underlying stochastic process follows a semi-Markov chain. In a case study on the VIX Index, efficient batch as well as sequential Bayesian parameter estimation schemes are contributed and validated. We benchmark HSMMs against popular discrete SSM alternatives and show how by-products that arise during the estimation process can be used for model selection and clustering. Chapter 3 centers on a new class of Susceptible-Exposed-Infected-Recovered (SEIR) type models to analyze and detect regime switches in the SARS-CoV-2 pandemic. We propose an epidemic model with the transmission rate between susceptible and infected individuals being time varying and piecewise constant. At any point in time, this parameter is linked to a latent variable that follows a HSMM. We define this model in state space formulation and demonstrate the latent states can be efficiently estimated using the Particle MCMC (PMCMC) and Sequential Monte Carlo Squared (SMC2) machinery. Moreover, a case study is conducted on the reported infection and fatalities data in the United Kingdom, during which we benchmark models with varying observation distribution specifications and determine the number of latent regimes in the data. Chapter 4 addresses Stochastic Volatility (SV) models and employs a variety of carefully selected copulas to explore the dependency structure between stocks and their volatility. This new class of models can reconstruct stylised empirical behaviours that cannot be captured by standard symmetric Gaussian innovations. In a case study on the S&P 500 and the VIX index, we examine the marginal distributions and joint dependency structure of the error terms in our proposed model. Moreover, batch and sequential Bayesian model selection are applied to analyze the suitability of the separate copula choices against standard modelling techniques.This thesis explores the topic of sequential inference on a variety of novel model classes. Chapter 2 focuses on a class of discrete State Space Models (SSM) known as Hidden Semi-Markov Model (HSMM), a versatile generalization of the famous Hidden Markov Model (HMM) in which the underlying stochastic process follows a semi-Markov chain. In a case study on the VIX Index, efficient batch as well as sequential Bayesian parameter estimation schemes are contributed and validated. We benchmark HSMMs against popular discrete SSM alternatives and show how by-products that arise during the estimation process can be used for model selection and clustering. Chapter 3 centers on a new class of Susceptible-Exposed-Infected-Recovered (SEIR) type models to analyze and detect regime switches in the SARS-CoV-2 pandemic. We propose an epidemic model with the transmission rate between susceptible and infected individuals being time varying and piecewise constant. At any point in time, this parameter is linked to a latent variable that follows a HSMM. We define this model in state space formulation and demonstrate the latent states can be efficiently estimated using the Particle MCMC (PMCMC) and Sequential Monte Carlo Squared (SMC2) machinery. Moreover, a case study is conducted on the reported infection and fatalities data in the United Kingdom, during which we benchmark models with varying observation distribution specifications and determine the number of latent regimes in the data. Chapter 4 addresses Stochastic Volatility (SV) models and employs a variety of carefully selected copulas to explore the dependency structure between stocks and their volatility. This new class of models can reconstruct stylised empirical behaviours that cannot be captured by standard symmetric Gaussian innovations. In a case study on the S&P 500 and the VIX index, we examine the marginal distributions and joint dependency structure of the error terms in our proposed model. Moreover, batch and sequential Bayesian model selection are applied to analyze the suitability of the separate copula choices against standard modelling techniques.