This paper discusses characteristics of standard conjugate priors and their induced posteriors in Bayesian inference for von Mises-Fisher distributions, using either the canonical natural exponential family or the more commonly employed polar coordinate parameterizations. We analyze when standard conjugate priors as well as posteriors are proper, and investigate the Jeffreys prior for the von Mises-Fisher family. Finally, we characterize the proper distributions in the standard conjugate family of the (matrixvalued) von Mises-Fisher distributions on Stiefel manifolds.

Orientador: Caio Lucidius Naberezny Azevedo / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-23T07:11:52Z (GMT). No. of bitstreams: 1 Nogarotto_DaniloCovaes_M.pdf: 12817108 bytes, checksum: 0e5e0de542d707f4023f5ef62dc40a82 (MD5) Previous issue date: 2013 / Resumo: No presente trabalho desenvolvemos ferramentas de inferência bayesiana para modelos de regressão beta e beta inflacionados, em relação à estimação paramétrica e diagnóstico. Trabalhamos com modelos de regressão beta não inflacionados, inflacionados em zero ou um e inflacionados em zero e um. Devido à impossibilidade de obtenção analítica das posteriores de interesse, tais ferramentas foram desenvolvidas através de algoritmos MCMC. Para os parâmetros da estrutura de regressão e para o parâmetro de precisão exploramos a utilização de prioris comumente empregadas em modelos de regressão, bem como prioris de Jeffreys e de Jeffreys sob independência. Para os parâmetros das componentes discretas, consideramos prioris conjugadas. Realizamos diversos estudos de simulação considerando algumas situações de interesse prático com o intuito de comparar as estimativas bayesianas com as frequentistas e também de estudar a sensibilidade dos modelos _a escolha de prioris. Um conjunto de dados da área psicométrica foi analisado para ilustrar o potencial do ferramental desenvolvido. Os resultados indicaram que há ganho ao se considerar modelos que contemplam as observações inflacionadas ao invés de transformá-las a fim de utilizar modelos não inflacionados / Abstract: In the present work we developed Bayesian tools, concerning parameter estimation and diagnostics, for noninflated, zero inflated, one inflated and zero-one inflated beta regression models. Due to the impossibility of obtaining the posterior distributions of interest, analytically, all these tools were developed through MCMC algorithms. For the regression and precision parameters we exploited the using of prior distributions commonly considered in regression models as well as Jeffreys and independence Jeffreys priors. For the parameters related to the discrete components, we considered conjugate prior distributions. We performed simulation studies, considering some situations of practical interest, in order to compare the Bayesian and frequentist estimates as well as to evaluate the sensitivity of the models to the prior choice. A psychometric real data set was analyzed to illustrate the performance of the developed tools. The results indicated that there is an overall improvement in using models that consider the inflated observations compared to transforming these observations in order to use noninflated models / Mestrado / Estatistica / Mestre em Estatística

Récemment, la grande complexité des applications modernes, par exemple dans la génétique, l’informatique, la finance, les sciences du climat, etc. a conduit à la proposition des nouveaux modèles qui peuvent décrire la réalité. Dans ces cas,méthodes MCMC classiques ne parviennent pas à rapprocher la distribution a posteriori, parce qu’ils sont trop lents pour étudier le space complet du paramètre. Nouveaux algorithmes ont été proposés pour gérer ces situations, où la fonction de vraisemblance est indisponible. Nous allons étudier nombreuses caractéristiques des modèles complexes: comment éliminer les paramètres de nuisance de l’analyse et faire inférence sur les quantités d’intérêt,dans un cadre bayésienne et non bayésienne et comment construire une distribution a priori de référence. / Recently, the great complexity of modern applications, for instance in genetics,computer science, finance, climatic science etc., has led to the proposal of newmodels which may realistically describe the reality. In these cases, classical MCMCmethods fail to approximate the posterior distribution, because they are too slow toinvestigate the full parameter space. New algorithms have been proposed to handlethese situations, where the likelihood function is unavailable. We will investigatemany features of complex models: how to eliminate the nuisance parameters fromthe analysis and make inference on key quantities of interest, both in a Bayesianand not Bayesian setting, and how to build a reference prior.

The dissertation considers construction of confidence intervals for a cumulative distribution function F(z) and its inverse at some fixed points z and u on the basis of an i.i.d. sample where the sample size is relatively small. The sample is modeled as having the flexible Generalized Gamma distribution with all three parameters being unknown. This approach can be viewed as an alternative to nonparametric techniques which do not specify distribution of X and lead to less efficient procedures. The confidence intervals are constructed by objective Bayesian methods and use the Jeffreys noninformative prior. Performance of the resulting confidence intervals is studied via Monte Carlo simulations and compared to the performance of nonparametric confidence intervals based on binomial proportion. In addition, techniques for change point detection are analyzed and further evaluated via Monte Carlo simulations. The effect of a change point on the interval estimators is studied both analytically and via Monte Carlo simulations.

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Research Doctorate - Doctor of Philosophy (PhD) / Interval estimation of the Binomial parameter è, representing the true probability of a success, is a problem of long standing in statistical inference. The landmark work is by Bayes (1763) who applied the uniform prior to derive the Beta posterior that is the normalised Binomial likelihood function. It is not well known that Bayes favoured this ‘noninformative’ prior as a result of considering the observable random variable x as opposed to the unknown parameter è, which is an important difference. In this thesis we develop additional arguments in favour of the uniform prior for estimation of è. We start by describing the frequentist and Bayesian approaches to interval estimation. It is well known that for common continuous models, while different in interpretation, frequentist and Bayesian intervals are often identical, which is directly related to the existence of a pivotal quantity. The Binomial model, and its Poisson sister also, lack a pivotal quantity, despite having sufficient statistics. Lack of a pivotal quantity is the reason why there is no consensus on one particular estimation method, more so than its discreteness: frequentist (unconditional) coverage depends on è. Exact methods guarantee minimum coverage to be at least equal to nominal and approximate methods aim for mean coverage to be close to nominal. We agree with what seems like the majority of frequentists, that exact methods are too conservative in practice, and show additional undesirable properties. This includes more recent ‘short’ exact intervals. We argue that Bayesian intervals based on noninformative priors are preferable to the family of frequentist approximate intervals, some of which are wider than exact intervals for particular data values. A particular property of the interval based on the uniform prior is that its mean coverage is exactly equal to nominal. However, once committed to the Bayesian approach there is no denying that the current preferred choice, by ‘objective’ Bayesians, is the U-shaped Jeffreys prior which results from various methods aimed at finding noninformative priors. The most successful such method seems to be reference analysis which has led to sensible priors in previously unsolved problems, concerning multiparameter models that include ‘nuisance’ parameters. However, we argue that there is a class of models for which the Jeffreys/reference prior may be suboptimal and that in the case of the Binomial distribution the requirement of a uniform prior predictive distribution leads to a more reasonable ‘consensus’ prior.