Bayesian perspectives on statistical modelling

This thesis explores the representation of probability measures in a coherent Bayesian modelling framework, together with the ensuing characterisation properties of posterior functionals. First, a decision theoretic approach is adopted to provide a unified modelling criterion applicable to assessing prior-likelihood combinations, design matrices, model dimensionality and choice of sample size. The utility structure and associated Bayes risk induces a distance measure, introducing concepts from differential geometry to aid in the interpretation of modelling characteristics. Secondly, analytical and approximate computations for the implementation of the Bayesian paradigm, based on the properties of the class of transformation models, are discussed. Finally, relationships between distance measures (in the form of either a derivative of a Bayes mapping or an induced distance) are explored, with particular reference to the construction of sensitivity measures

Bayesian perspectives on statistical modelling

This thesis explores the representation of probability measures in a coherent Bayesian modelling framework, together with the ensuing characterisation properties of posterior functionals. First, a decision theoretic approach is adopted to provide a unified modelling criterion applicable to assessing prior-likelihood combinations, design matrices, model dimensionality and choice of sample size. The utility structure and associated Bayes risk induces a distance measure, introducing concepts from differential geometry to aid in the interpretation of modelling characteristics. Secondly, analytical and approximate computations for the implementation of the Bayesian paradigm, based on the properties of the class of transformation models, are discussed. Finally, relationships between distance measures (in the form of either a derivative of a Bayes mapping or an induced distance) are explored, with particular reference to the construction of sensitivity measures

Applications of differential geometry to statistics

Chapters 1 and 2 are both surveys of the current work in applying geometry to statistics. Chapter 1 is a broad outline of all the work done so far, while Chapter 2 studies, in particular, the work of Amari and that of Lauritzen. In Chapters 3 and 4 we study some open problems which have been raised by Lauritzen's work. In particular we look in detail at some of the differential geometric theory behind Lauritzen's defmition of a Statistical manifold. The following chapters follow a different line of research. We look at a new non symmetric differential geometric structure which we call a preferred point manifold. We show how this structure encompasses the work of Amari and Lauritzen, and how it points the way to many generalizations of their results. In Chapter 5 we define this new structure, and compare it to the Statistical manifold theory. Chapter 6 develops some examples of the new geometry in a statistical context. Chapter 7 starts the development of the pure theory of these preferred point manifolds. In Chapter 8 we outline possible paths of research in which the new geometry may be applied to statistical theory. We include, in an appendix, a copy of a joint paper which looks at some direct applications of differential geometry to a statistical problem, in this case it is the problem of the behaviour of the Wald test with nonlinear restriction functions

Recherche d'images par le contenu, analyse multirésolution et modèles de régression logistique

Cette thèse, présente l'ensemble de nos contributions relatives à la recherche d'images par le contenu à l'aide de l'analyse multirésolution ainsi qu'à la classification linéaire et nonlinéaire. Dans la première partie, nous proposons une méthode simple et rapide de recherche d'images par le contenu. Pour représenter les images couleurs, nous introduisons de nouveaux descripteurs de caractéristiques qui sont des histogrammes pondérés par le gradient multispectral. Afin de mesurer le degré de similarité entre deux images d'une façon rapide et efficace, nous utilisons une pseudo-métrique pondérée qui utilise la décomposition en ondelettes et la compression des histogrammes extraits des images. Les poids de la pseudo-métrique sont ajustés à l'aide du modèle classique de régression logistique afin d'améliorer sa capacité à discriminer et la précision de la recherche. Dans la deuxième partie, nous proposons un nouveau modèle bayésien de régression logistique fondé sur une méthode variationnelle. Une comparaison de ce nouveau modèle au modèle classique de régression logistique est effectuée dans le cadre de la recherche d'images. Nous illustrons par la suite que le modèle bayésien permet par rapport au modèle classique une amélioration notoire de la capacité à discriminer de la pseudo-métrique et de la précision de recherche. Dans la troisième partie, nous détaillons la dérivation du nouveau modèle bayésien de régression logistique fondé sur une méthode variationnelle et nous comparons ce modèle au modèle classique de régression logistique ainsi qu'à d'autres classificateurs linéaires présents dans la littérature. Nous comparons par la suite, notre méthode de recherche, utilisant le modèle bayésien de régression logistique, à d'autres méthodes de recherches déjà publiées. Dans la quatrième partie, nous introduisons la sélection des caractéristiques pour améliorer notre méthode de recherche utilisant le modèle introduit ci-dessus. En effet, la sélection des caractéristiques permet de donner automatiquement plus d'importance aux caractéristiques qui discriminent le plus et moins d'importance aux caractéristiques qui discriminent le moins. Finalement, dans la cinquième partie, nous proposons un nouveau modèle bayésien d'analyse discriminante logistique construit à l'aide de noyaux permettant ainsi une classification nonlinéaire flexible

Bayesian inference for quantiles and conditional means in log-normal models

The main topic of the thesis is the proper execution of a Bayesian inference if log-normality is assumed for data. In fact, it is known that a particular care is required in this context, since the most common prior distributions for the variance in log scale produce posteriors for the log-normal mean which do not have finite moments. Hence, classical summary measures of the posterior such as expectation and variance cannot be computed for these distributions. The thesis is aimed at proposing solutions to carry out Bayesian inference inside a mathematically coherent framework, focusing on the estimation of two quantities: log-normal quantiles (first part of the thesis) and conditioned expectations under a general log-normal linear mixed model (second part of the thesis). Moreover, in the latter section, a further investigation on a unit-level small area models is presented, considering the problem of estimating the well-known log-transformed Battese, Harter and Fuller model in the hierarchical Bayes context. Once the existence conditions for the moments of the target functionals posterior are proved, new strategies to specify prior distributions are suggested. Then, the frequentist properties of the deduced Bayes estimators and credible intervals are evaluated through accurate simulations studies: it resulted that the proposed methodologies improve the Bayesian estimates under naive prior settings and are satisfactorily competitive with the frequentist solutions available in the literature. To conclude, applications of the developed inferential strategies are illustrated on real datasets. The work is completed by the implementation of an R package named BayesLN which allows the users to easily carry out Bayesian inference for log-normal data

Health outcomes and income inequality : a multilevel analysis of the Wilkinson hypothesis

EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Investigating the spatial distribution of diabetes in Africa using both classical and Bayesian approaches.

Master of Science in Statistics. University of KwaZulu-Natal, Durban, 2017.Abstract available in PDF file

'Visions of an unseen world': the production and consumption of English ghost stories, c.1660-1800

This thesis traces the cultural significance of ghost beliefs in English society from c.1660 to c.1800. It is an attempt to partially re-enchant these years and to nuance historical characterisation of eighteenth-century England as an enlightened, secularising and ‘anti-superstitious’ nation. Moreover, I aim to restore ghost beliefs to historical legitimacy and my central argument is that they played a crucial role in shaping the specific social, political, economic and religious contours of eighteenth-century life. Ghosts have been largely exorcised from existing accounts of this period and so this research represents a fresh contribution to historical understandings of the long eighteenth century and to historiographies of the supernatural more generally. The following chapters describe how ghost beliefs blended with the religious cultures of Anglicanism and Methodism by reinforcing orthodox theological teachings. The idea that dead souls could return to earth also complemented clerical initiatives to reform lay spirituality and to temper the extremes of rational religion. I chart how ghost beliefs fared in the face of new enlightenment philosophies, and how they informed discourse of politeness, individuality and interiority. This is accompanied by explorations of the relevance of ghost beliefs in everyday life. I describe the places and spaces in which ghost stories were told, the people who narrated them and those who listened. This ‘thick description’ emphasises how the spread of ghost stories was encouraged by contemporary labour relations, by the expansion of British imperial and trading interests overseas, and by patterns of sociability that were intrinsically linked to the realities of eighteenth-century life. I have harnessed insights from socio-linguistics and the sociology of literature to theorise the relationship between ghost stories and ghost beliefs. I have examined the production, circulation and consumption of ghost stories, as well as their form and content, to explain how these texts reflected and shaped the opinions of a variety of readers. In so doing, this thesis suggests an important relationship between literary forms and historical change

Bayesian spatio-temporal modelling of rainfall through non-homogenous hidden Markov models

Multi-site statistical models for daily rainfall should account for spatial and temporal dependence amongst measurements and also allow for the event of no rain. Recent research into climate change and variability has sparked interest in the relationship between rainfall and climate, stimulating the development of statistical models that relate large-scale atmospheric variables to local precipitation. Although modelling daily rainfall presents a challenging and topical problem, there have been few attempts taking a subjective Bayesian approach. This thesis is concerned with developing hidden Markov models (HMMs) for the spatio-temporal analysis of rainfall data, within a Bayesian framework. In these models, daily rainfall patterns are driven by a finite number of unobserved states, interpreted as weather states, that evolve in time as a first order Markov chain. The weather states explain space time structure in the data so that reasonably simple models can be adopted within states. Throughout this thesis, the models and procedures are illustrated using data from a small dense network of six sites situated in Yorkshire, UK. First we study a simple (homogeneous) HMM in which rainfall occurrences and amounts, given occurrences, are conditionally independent in space and time, given the weather state, and have Bernoulli and gamma distributions, respectively. We compare methods for approximating the posterior distribution for the number of weather states. This simple model does not incorporate atmospheric information and appears not to capture the observed spatio-temporal structure. We therefore investigate two non-homogeneous hidden Markov models (NHMMs) in which we allow the transition probabilities between weather states to depend on time-varying atmospheric variables and successively relax the conditional independence assumptions. The first NHMM retains the simple conditional model for non-zero rainfall amounts but allows occurrences to form a Markov chain of autologistic models, given the weather state. The second introduces latent multivariate normal random variables to form a hierarchical NHMM in which neither rainfall occurrences nor non-zero amounts are conditionally spatially or temporally independent, given the weather state. Throughout this thesis, we emphasise the elicitation of prior distributions that convey genuine initial beliefs. For each hidden Markov model studied we demonstrate techniques to assist in this task.EThOS - Electronic Theses Online ServiceEngineering and Physical Sciences Research CouncilGBUnited Kingdo