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This thesis is concerned with the development of Bayesian methods for inference and deconvolution. We compare and contrast different Bayesian methods for model selection, specifically Markov Chain Monte Carlo methods (MCMC) and Variational methods and their application to medical and industrial problems. In chapter 1, the Bayesian framework is outlined. In chapter 2 the different methods for Bayesian model selection are introduced and we assess each method in turn. Problems with MCMC methods and Variational methods are highlighted, before a new method which combines the strengths of both the MCMC methods and the Variational methods is developed. Chapter 3 applies the inferential methods described in chapter 2 to the problem of interpolation, before a regression neural network is implemented and tested on a set of data from the microelectronics industry. Chapter 4 applies the interpolation methods developed in chapter three to characterise the electrical nature of the testing site in the integrated circuit (IC) manufacturing process. Chapter 5 describes Independent Component Analysis (ICA) as a solution to the bilinear decomposition problem and its application to Magnetic Resonance Imaging. This chapter also compares and contrasts various Bayesian algorithms for the bilinear problem with a non-Bayesian MUSIC algorithm. Chapter 6 describes various models for the deconvolution of images including a regression network. The ICA model of chapter 5 is then extended to the deconvolution and blind deconvolution problems with the addition of intrinsic correlation functions.

This dissertation is concerned with various mathematical aspects of Topological Quantum Field Theories (TQFTs) known as Chern-Simons theories. Although this subject has its origins in theoretical physics, the treatment here is in terms of the axiomatic approach due to Segal and Atiyah. A key feature of the thesis is the notion of a 3-tier (axiomatic) TQFT. This involves assigning a category to a closed I-manifold and a functor to a 2-manifold with boundary which is viewed as a cobordism between I-manifolds. To a closed 2-manifold Σ the theory assigns a vector space <I>H<SUB>Σ</SUB></I> , and to a 3-manifold <I>M</I> the theory assigns a numerical invariant (if <I>M </I>is closed), a vector in <I>H<SUB>δM</SUB></I> (if <I>M</I> has closed boundary <I>δM</I> ) or a natural transformation of functors (if the boundary <I>δM</I> of <I>M</I> has a 1-dimensional boundary). After a brief introduction, we introduce in chapter 1 the definition of a TQFT and that of a 3-tier TQFT. We then describe the geometrical set-up for Chern-Simons Theory for a Lie group <I>G </I>and focus on the particular case of <I>G = SU</I>(2). Finally we describe quite concisely how it might fit into a 3-tier TQFT structure. Roughly the next half of the thesis treats the specific case of Chern-Simons theory for the circle group T. In chapter 2 we describe a number of interesting topological aspects of the theory. In chapter 3 we go on to show how the theory fits into 3-tier TQFT framework. In the next two chapters we begin to deal with Chern-Simons theories for <I>G </I>a non-compact group. In chapters 6 and 7 we deal with a rather more algebraic theory which is the abelian version of a theory which is meant to compute the Casson invariant for oriented homology 3-spheres. For this reason we call it the abelian Casson-type theory. From the physics viewpoint, it coincides with the Chern-Simons theory where <I>G</I> is a supergroup. This is rather difficult to motivate mathematically, so we adopt an algebraic-topological definition of the theory and show it satisfies the TQFT axioms. We then go on to show how it fits into the 3-tier TQFT structure. The novelty here is that the category assigned to a 1-manifold is not semisimple.