Bayesian Non-Parametric Inference for Infectious Disease Data

We propose a framework for Bayesian non-parametric estimation of the rate at which new infections occur assuming that the epidemic is partially observed. The developed methodology relies on modelling the rate at which new infections occur as a function which only depends on time. Two different types of prior distributions are proposed namely using step-functions and B-splines. The methodology is illustrated using both simulated and real datasets and we show that certain aspects of the epidemic such as seasonality and super-spreading events are picked up without having to explicitly incorporate them into a parametric model

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 effcient MCMC algorithms than the centered ones. The methodology developed for drawing eciently 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 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 fndings 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

A two-level Markov model for packet loss in UDP/IP-based real-time video applications targeting residential users

The packet loss characteristics of Internet paths that include residential broadband links are not well understood, and there are no good models for their behaviour. This compli- cates the design of real-time video applications targeting home users, since it is difficult to choose appropriate error correction and concealment algorithms without a good model for the types of loss observed. Using measurements of residential broadband networks in the UK and Finland, we show that existing models for packet loss, such as the Gilbert model and simple hidden Markov models, do not effectively model the loss patterns seen in this environment. We present a new two-level Markov model for packet loss that can more accurately describe the characteristics of these links, and quantify the effectiveness of this model. We demonstrate that our new packet loss model allows for improved application design, by using it to model the performance of forward error correction on such links

Exact Bayesian inference via data augmentation

Data augmentation is a common tool in Bayesian statistics, especially in the application of MCMC. Data augmentation is used where direct computation of the posterior density, π(θ|x), of the parameters θ, given the observed data x, is not possible. We show that for a range of problems, it is possible to augment the data by y, such that, π(θ|x,y) is known, and π(y|x) can easily be computed. In particular, π(y|x) is obtained by collapsing π(y,θ|x) through integrating out θ. This allows the exact computation of π(θ|x) as a mixture distribution without recourse to approximating methods such as MCMC. Useful byproducts of the exact posterior distribution are the marginal likelihood of the model and the exact predictive distribution