Efficient Bayesian Inference for Partially Observed Stochastic Epidemics and A New Class of Semi−Parametric Time Series Models

This thesis is divided in two distinct parts. In the first part we are concerned with developing new statistical methodology for drawing Bayesian inference for partially observed stochastic epidemic models. In the second part, we develop a novel methodology for constructing a wide class of semi−parametric time series models. First, we introduce a general framework for the heterogeneously mixing stochastic epidemic models (HMSE) and we also review some of the existing methods of statistical inference for epidemic models. The performance of a variety of centered Markov Chain Monte Carlo (MCMC) algorithms is studied. It is found that as the number of infected individuals increases, then the performance of these algorithms deteriorates. We then develop a variety of centered, non−centered and partially non−centered reparameterisations. We show that partially non−centered reparameterisations often offer more efficient MCMC algorithms than the centered ones. The methodology developed for drawing efficiently Bayesian inference for HMSE is then applied to the 2001 UK Foot-and-Mouth disease outbreak in Cumbria. Unlike other existing modelling approaches, we model stochastically the infectious period of each farm assuming that the infection date of each farm is typically III unknown. Due to the high dimensionality of the problem, standard MCMC algorithms are inefficient. Therefore, a partially non−centered algorithm is applied for the purpose of obtaining reliable estimates for the model’s parameter of interest. In addition, we discuss similarities and differences of our findings in comparison to other results in the literature. The main purpose of the second part of this thesis, is to develop a novel class of semi−parametric time series models. We are interested in constructing models for which we can specify in advance the marginal distribution of the observations and then build the dependence structure of the observations around them. First, we review current work concerning modelling time series with fixed non−Gaussian margins and various correlation structures. Then, we introduce a stochastic process which we term a latent branching tree (LBT). The LBT enables us to allow for a rich variety of correlation structures. Apart from discussing in detail the tree’s properties, we also show how Bayesian inference can be carried out via MCMC methods. Various MCMC strategies are discussed including non−centered parameterisations. It is found that non−centered algorithms significantly improve the mixing of some of the algorithms based on centered reparameterisations. Finally, we present an application of this class of models to a real dataset on genome scheme data.