This thesis considers the development of efficient MCMC sampling methods for Bayesian models used for the pairwise alignment of two unlabelled configurations. We introduce ideas from differential geometry along with other recent developments in unlabelled shape analysis as a means of creating novel and more efficient MCMC sampling methods for such models. For example, we have improved the performance of the sampler for the model of Green and Mardia (2006) by sampling rotation, A ∈ SO(3), and matching matrix using geodesic Monte Carlo (MCMC defined on manifold) and Forbes and Lauritzen (2014) matching sampler, developed for finger print matching problem, respectively. We also propose a new Bayesian model, together with implementation methods, motivated by the desire for further improvement. The model and its implementation methods proposed exploit the continuous nature of the parameter space of our Bayesian model and thus move around easily in this continuous space, providing highly efficient convergence and exploration of the target posterior distribution. The proposed Bayesian model and its implementation methods provide generalizations of the existing two models, Bayesian Hierarchical and regression models, introduced by Green and Mardia (2006) and Taylor, Mardia and Kent (2003) respectively, and resolve many shortcomings of existing implementation methods; slow convergence, traps in local mode and dependence on initial starting values when sampling from high dimensional and multi-modal posterior distributions. We illustrate our model and its implementation methods on the alignment of two proteins and two gels, and we find that the performance of proposed implementation methods under proposed model is better than current implementation techniques of existing models in both real and simulated data sets.

Peng Heng. / "June 2003." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (p. 93-100). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Wong Wai-Wan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 45-46). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Ipsative Data --- p.1 / Chapter 1.2 --- Transformation --- p.2 / Chapter 1.3 --- Influence Analysis --- p.3 / Chapter 2 --- Ipsative Data --- p.5 / Chapter 2.1 --- Additive Ipsative Data (AID) and Multiplicative Ipsative Data (MID) --- p.6 / Chapter 2.1.1 --- Additive Ipsative Data (AID) --- p.6 / Chapter 2.1.2 --- Multiplicative Ipsative Data (MID) --- p.7 / Chapter 2.2 --- Partially Additive Ipsative Data (PAID) --- p.7 / Chapter 2.2.1 --- Vector of PAID --- p.8 / Chapter 2.2.2 --- Special Cases of PAID --- p.9 / Chapter 3 --- Transformation --- p.10 / Chapter 3.1 --- Distribution of AID --- p.10 / Chapter 3.2 --- Transformation --- p.11 / Chapter 3.3 --- Relationship between Parameters of AID and the Transformed Vector --- p.13 / Chapter 4 --- Influence Analysis of Ipsative Data --- p.14 / Chapter 4.1 --- The Postulated Model --- p.15 / Chapter 4.2 --- Perturbation --- p.16 / Chapter 4.3 --- Likelihood Displacement --- p.17 / Chapter 4.4 --- Normal Curvature --- p.18 / Chapter 4.5 --- Computation of the Normal Curvature --- p.20 / Chapter 4.6 --- Diagnostic Measures --- p.21 / Chapter 4.6.1 --- Observations influencing the estimates of μ and Σ --- p.22 / Chapter 4.6.2 --- Observations influencing the estimate of μ or Σ --- p.22 / Chapter 4.7 --- Examples --- p.24 / Chapter 4.7.1 --- Example 1: Foraminiferal Compositions Data Set (AID) . --- p.24 / Chapter 4.7.2 --- Example 2: Compositions of Sediments Data Set (PAID) . --- p.25 / Chapter 5 --- Discussion --- p.31 / Chapter A --- Proof of Propositions --- p.34 / Chapter A.1 --- Proof of Proposition 3 --- p.34 / Chapter A.2 --- Proof of Proposition 4 --- p.35 / Chapter B --- "Analytical Expressions of (Lθ*)-1, Δ* and Lc" --- p.37 / Chapter B.1 --- Analytical Expression of (Lθ*)-1 --- p.37 / Chapter B.2 --- Analytical Expression of Δ* --- p.38 / Chapter B.3 --- Analytical Expression of Lc --- p.38 / Chapter C --- Calculation of aθ- --- p.39 / Chapter D --- Matlab Commands of Example 1 --- p.42 / Bibliography --- p.44

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Modelling of infectious diseases is a topic of great importance. Despite the enormous attention given to the development of methods for efficient parameter estimation, there has been relatively little activity in the area of nonparametric inference for epidemics. In this thesis, we develop new methodology which enables nonparametric estimation of the parameters which govern transmission within a Bayesian framework. Many standard modelling and data analysis methods use underlying assumptions (e.g. concerning the rate at which new cases of disease will occur) which are rarely challenged or tested in practice. We relax these assumptions and analyse data from disease outbreaks in a Bayesian nonparametric framework. We first apply our Bayesian nonparametric methods to small-scale epidemics. In a standard SIR model, the overall force of infection is assumed to have a parametric form. We relax this assumption and treat it as a function which only depends on time. Then we place a Gaussian process prior on it and infer it using data-augmented Markov Chain Monte Carlo (MCMC) algorithms. Our methods are illustrated by applications to simulated data as well as Smallpox data. We also investigate the infection rate in the SIR model using our methods. More precisely, we assume the infection rate is time-varying and place a Gaussian process prior on it. Results are obtained using data augmentation methods and standard MCMC algorithms. We illustrate our methods using simulated data and respiratory disease data. We find our methods work fairly well for the stochastic SIR model. We also investigate large-scaled epidemics in a Bayesian nonparametric framework. For large epidemics in large populations, we usually observe surveillance data which typically provide number of new infection cases occurring during observation periods. We infer the infection rate for each observation period by placing Gaussian process priors on them. Our methods are illustrated by the real data, i.e. a time series of incidence of measles in London (1948-1957). Please note, the pagination in the online version differs slightly from the official, printed version because of the insertion of a list of corrections. The incorporation of the corrections into the text of the online version means that the page breaks appear at different points on p. 39-47, and p. 47-147 of the electronic version correspond to p. 48-148 of the printed version.

Errate slip inserted. Bibliography: leaves 130-134.

Thesis (Ph.D.)--McMaster University, 2005. / Supervisor: N. Balakrishnan. Includes bibliographical references (p. 148-152).

Thesis (M.S.)--Oregon State University, 1965. / Typescript. Includes bibliographical references (leaves 34-35). Also available on the World Wide Web.

<p>This thesis consists of four papers.</p><p>In paper 1, we prove central limit theorems for Markov chains under (local) contraction conditions. As a corollary we obtain a central limit theorem for Markov chains associated with iterated function systems with contractive maps and place-dependent Dini-continuous probabilities.</p><p>In paper 2, properties of inverse subordinators are investigated, in particular similarities with renewal processes. The main tool is a theorem on processes that are both renewal and Cox processes.</p><p>In paper 3, distributional properties of supercritical and especially immortal branching processes are derived. The marginal distributions of immortal branching processes are found to be compound geometric.</p><p>In paper 4, a description of a dynamic population model is presented, such that samples from the population have genealogies as given by a Lambda-coalescent with mutations. Depending on whether the sample is grouped according to litters or families, the sampling distribution is either regenerative or non-regenerative.</p>