A major problem associated with Bayesian estimation is selecting the prior distribution. The more recent literature on the selection of the prior is reviewed. Very little of a general nature on the selection of the prior is formed in the literature except for non-informative priors. This class of priors is seen to have limited usefulness. A method of selecting an informative prior is generalized in this thesis to include estimation of several parameters using a multivariate prior distribution. The concepts required for quantifying prior information is based on intuitive principles. In this way, it can be understood and controlled by the decision maker (i.e., those responsible for the consequences) rather than analysts. The information required is: (1) prior point estimates of the parameters being estimated and (2) an expression of the desired influence of the prior relative to the present data in determining the parameter estimates (e.g., item (2) implies twice as much influence as the data). These concepts (point estimates and influence) may be used equally with subjective or quantitative prior information.

In the master thesis the issue of bayesian approach to stochastic reserving is solved. Reserving problem is very discussed in insurance industry. The text introduces the basic actuarial notation and terminology and explains the bayesian inference in statistics and estimation. The main part of the thesis is framed by the description of the particular bayesian models. It is focused on the derivation of estimators for the reserves and ultimate claims. The aim of the thesis is to show the practical uses of the models and the relations between them. For this purpose the methods are applied on a real data set. Obtained results are summarized in tables and the comparison of the methods is provided. Finally the impact of a prior distribution on the resulting reserves is showed. Powered by TCPDF (www.tcpdf.org)

In this thesis channel capacity of a special type of multiple-input multiple-output (MIMO) Rayleigh fading channels is studied, where the transmitters are subject to a finite phase-shift keying (PSK) input alphabet. The constraint on the input alphabet makes an analytical solution for the capacity beyond reach. However we are able to simplify the final expression, which requires a single expectation and thus can be evaluated easily through simulation. To facilitate simulations, analytical expressions are derived for the eigenvalues and eigenvectors of a covariance matrix involved in the simplified capacity expression. The simplified expression is used to provide some good approximations to the capacity at low signal-to-noise ratios (SNRs). Involved in derivation of the capacity is the capacity-achieving input distribution. It is proved that a uniform prior distribution is capacity achieving. We also show that it is the only capacity-achieving distribution for our channel model. On top of that we generalize the uniqueness case for an input distribution to a broader range of channels.

Estimation of location and scale parameters from a random sample of size n is of paramount importance in Statistics. An estimator is called fully efficient if it attains the Cramer-Rao minimum variance bound besides being unbiased. The method that yields such estimators, at any rate for large n, is the method of modified maximum likelihood estimation. Apparently, such estimators cannot be made more efficient by using sample based classical methods. That makes room for Bayesian method of estimation which engages prior distributions and likelihood functions. A formal combination of the prior knowledge and the sample information is called posterior distribution. The posterior distribution is maximized with respect to the unknown parameter(s). That gives HPD (highest probability density) estimator(s). Locating the maximum of the posterior distribution is, however, enormously difficult (computationally and analytically) in most situations. To alleviate these difficulties, we use modified likelihood function in the posterior distribution instead of the likelihood function. We derived the HPD estimators of location and scale parameters of distributions in the family of Generalized Logistic. We have extended the work to experimental design, one way ANOVA. We have obtained the HPD estimators of the block effects and the scale parameter (in the distribution of errors) / they have beautiful algebraic forms. We have shown that they are highly efficient. We have given real life examples to illustrate the usefulness of our results. Thus, the enormous computational and analytical difficulties with the traditional Bayesian method of estimation are circumvented at any rate in the context of experimental design.

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L'estimation des abondances de population est essentielle pour comprendre les dynamiques de population, les interactions entre espèces et estimer les risques de transmission d'agents pathogènes dans les populations. Plusieurs méthodes d'échantillonnages, basées sur des hypothèses spécifiques permettent d'estimer ces abondances : les méthodes par comptages uniques, par « distance sampling », par échantillonnages successifs ou par capture marquage recapture. Nous nous sommes intéressés à l'abondance des tiques Ixodes ricinus, vecteurs de nombreux agents pathogènes. Cette abondance est classiquement estimée par le nombre de tiques capturées lors d'échantillonnages uniques réalisés sur différentes unités d'observation. Cependant, de nombreuses études remettent en cause cette hypothèse forte et suggèrent que le taux d'échantillonnage est variable selon les conditions d'échantillonnage (type de végétation,…) mais ne prennent pas en compte ce taux d'échantillonnage pour autant. A partir d'une méthode d'échantillonnage par « removal sampling » (RS), (i) nous avons montré que les conditions environnementales influençaient le taux d'échantillonnage et l'indicateur d'abondance usuel i.e. le nombre de tiques capturées lors d'un seul échantillonnage (ii) nous avons proposé une méthode pour détecter l'indicateur d'abondance, basés sur le nombre cumulé de capture, le moins soumis aux variations du taux ; (iii) par une approche Bayésienne hiérarchique, nous avons estimé simultanément l'abondance de tiques des unités d'observation et la valeur du taux d'échantillonnage en fonction du type de végétation et de l'heure d'échantillonnage. Nous avons montré que le taux d'échantillonnage sur des arbustes (entre 33,9 % et 47,4%) était significativement inférieur au taux d'échantillonnage sur des feuilles mortes (entre 53,6 % et 66,7%). De plus, nous avons montré que le modèle RS tend vers un modèle de Poisson iid lorsque la taille de la population N0 tend vers l'infini ce qui pose des problèmes d'indétermination pour estimer les paramètres N0 et τ, le taux d'échantillonnage. Nous avons également montré que (i) les estimateurs Bayésiens divergent lorsque les lois a priori sont des lois vagues ; (ii) les lois a priori β(a, b) avec a > 2 sur τ conduisaient à des estimateurs Bayésien convergents. Enfin, nous avons proposé des recommandations quant au choix des lois a priori pour τ afin d'obtenir de bonnes estimations pour N0 ou pour τ. Nous discutons de la pertinence des méthodes RS pour les tiques et des perspectives envisageables pour (i) estimer le risque acarologique représenté par la population de tiques potentiellement actives sur une unité d'observation, (ii) estimer un risque à l'échelle d'une parcelle, à savoir comment répartir l'effort d'échantillonnage entre le nombre d'unités d'observation et le nombre d'échantillonnages successifs par unités d'observation. / The estimation of animal abundance is essential to understand population dynamics, species interactions and disease patterns in populations and to estimate the risk of pathogens transmission. Several sampling methods such as single counts, distance sampling, removal sampling or capture mark recapture could be used to estimate abundance. In this study, we are investigated the abundance of Ixodes ricinus ticks, which are involved in the transmission of many pathogens. Tick abundance is commonly estimated by the number of nymphs captured during a single observation (a cloth dragged on a given surface). In this case, analyses of abundance patterns assumes that the probability of detecting a tick, hence the sampling rate, remains constant across the observations. In practice, however, this assumption is often not satisfied as the sampling rate may fluctuate between observation plots. The variation of sampling rate is never taken into account in estimations of tick abundance. Using a removal sampling design (RS), (i) we showed that the sampling rate and the usual abundance indicator (based on a single drag observation per spot) were both influenced by environmental conditions ; (ii) we proposed a method to determine the abundance indicator the least influenced by sampling rate variations ; (iii) using a hierarchical Bayesian model, we estimated simultaneously the abundance and the sampling rate according the type of vegetation, and the time of sampling. The sampling rate varied between 33,9 % and 47,4 % for shrubs and 53,6 % and 66,7 % for dead leaves. In addition, we show that the RS model tends to Poisson iid model when the population size N0 tends to infinite. This result conduct to infinite estimations for N0. We show that (i) Bayesian estimators were divergent for vague prior ; (ii) β(a, b) prior for a > 2 on τ conduct to convergent estimators. Then, we proposed recommendations for prior choice for τ parameter to give good estimations of N0 or τ. We discuss the relevance of RS for ticks and the possible perspectives to (i) estimate the acarologic risk associated to all potential active ticks for given spot, (ii) estimate the risk at the larger scale, i.e. how to distribute the sampling effort between number of spot and number of consecutive sampling by spot.

This thesis assesses the use of the measured motions of a semisubmersible oil platform as a basis for estimating on-site wave spectra. The inference method followed is based on the wave buoy analogy, which aims at solving the linear inverse problem: estimate the sea state, given the measured motions and the transfer function of the platform. Directional wave inference obtained from the records of vessels motions is a technique that has seen its application grow signicantly over the last years. As a matter of fact, its applications in ships with forward speed and ship-shaped moored platforms (such as FPSOs) have provided good results. However, little research has been done regarding the use of semisubmersible platforms as wave sensors. This is due to the fact that these platforms are designed to present no signicant responses when excited by waves. Notwithstanding this, the semisubmersible platforms are characterized by measurable small motions. Moreover, if compared with ship-shaped motion-based wave sensors, the responses of the semisubmersibles are in better agreement with the response characteristics estimations obtained by means of linear hydrodynamic models. In addition, the eminently linear characteristics of the responses often lasts even for severe wave conditions. This feature results in that the semisubmersible platforms stand as a promising wave sensor even for extreme sea states, conditions in which other types of sensors (i.e. buoys, radars) may face diculties. Throughout the text, the main results of this work are presented and discussed. These results are mainly based on a dedicated experimental campaign, carried out with a scaled model of the Asgar-B platform, which is a semisubmersible platform located in the Asgard eld oshore Norway. Regarding the sea states tested during the experiential campaign, they were estimated by means of a motion-based Bayesian inference method, which has been developed for more than then years at the EPUSP. In order to allow the adoption of the semisubmersible platforms as a motion based wave sensors, this thesis provides two signicant improvements of the method: rst, a method to obtain an estimation of the linearized equivalent external viscous damping is provided. This analytical methodology allows to reduce the uncertainty of the transfer function of the platform close to the resonances of the motions and, as a consequence, it increases the accuracy of the inference approach. The second relevant contribution is the development of an alternative prior distribution, which is adopted to introduce the prior beliefs regarding the sea state in the Bayesian inference approach. It is shown that although some aspects of this novel approach require further evaluation in future work, the prior distribution developed has potential to improve the accuracy of wave estimates, and, at the same time, it signi cantly simplies the calibration procedures followed by other state-of-the-art Bayesian wave inference methods. Summing up, the inference approach proposed in this work provides the bases to use each semisubmersible oil platform, which stand as the most common type of oil platforms operated oshore Brasil, as a motion based wave sensor, thus contributing to the possible broadening of the Brazilian oceanographic measurement network. / A presente tese investiga a adoção de plataformas de petróleo semi submersíveis como base para inferência das condições de onda através do monitoramento de seus movimentos. O problema em questão consiste na solução do problema inverso de comportamento em ondas; ou seja, uma vez observados os movimentos da unidade flutuante (e conhecidas suas funções de resposta de movimento), estima-se as condições de ondas que os causaram. Este tipo de método já vem sendo empregado há anos para navios em curso e também para navios convertidos em plataformas de petróleo (os chamados FPSOs) com bons resultados. No entanto, o possível emprego de plataformas semi-submersíveis para o mesmo fim foi muito pouco explorado até o momento. Evidentemente, isso decorre da suposição de que, uma vez que essas estruturas são projetadas com o intuito primeiro de atenuar os movimentos decorrentes das ações de ondas, naturalmente elas não seriam bons sensores para esta finalidade. Os resultados apresentados nesta tese, todavia, contrariam tal suposição. De fato, as semi-submersíveis respondem de forma fraca as ondas, porem esta resposta é mensurável. Não apenas isso, mas, em comparação com os cascos de navios, esta resposta adere melhor às previsões dos modelos hidrodinâmicos lineares a partir dos quais as características da plataforma são estimadas. Ademais, o caráter eminentemente linear da resposta muitas vezes perdura inclusive para condições de ondas severas. Isto, por sua vez, torna as semi-submersíveis promissoras inclusive para a estimação de mares extremos, situação nas quais os outros tipos de sensores (boias, radares) enfrentam dificuldades. Nesta tese, a demonstração destes fatos é sustentada por um extenso conjunto de testes experimentais realizados em tanque de ondas com um modelo em escala reduzida de uma plataforma que hoje opera no Mar do Norte. Para tanto, foi empregado um método de inferência Bayesiana para estimação de ondas em navios que vem sendo desenvolvido na EPUSP há mais de dez anos. Para o estudo das semi-submersíveis o trabalho propõe duas melhorias importantes no método: A primeira consiste em um procedimento analítico para prever o amortecimento hidrodinâmico de origem viscosa dos movimentos observados do casco. Este procedimento permite reduzir as incertezas quanto a função de resposta em condições de ressonância dos movimentos com as ondas e, dessa forma, aumentar a confiabilidade do método. A segunda contribuição relevante é a proposição de uma alternativa para a chamada distribuição a priori originalmente empregada pelo método Bayesiano. Demonstra-se que, embora alguns aspectos desta nova metodologia ainda necessitem de uma avaliação adicional em trabalhos futuros, a nova distribuição tem grande potencial para melhorar a precisão das estimativas de ondas, além de simplificar de maneira significativa os procedimentos atuais de calibração do sistema de inferência. Em suma, o método de inferência aqui proposto abre caminho para tornar cada unidade flutuante de óleo e gás do tipo semi-submersível, um dos sistemas de produção mais frequentes nas costas brasileiras, um eventual ponto de monitoramento de ondas, contribuindo então para a possível ampliação de nossas bases de medição oceanograficas.

L'approche Bayésienne permet d’enrichir l'information apportée par l'essai clinique, en intégrant des informations externes à l'essai. De plus, elle permet d’exprimer les résultats directement en termes de probabilité d’un certain effet du traitement, plus informative et interprétable qu’une p-valeur et un intervalle de confiance. Par ailleurs, la réduction fréquente d’une analyse à une interprétation binaire des résultats (significatif ou non) est particulièrement dommageable dans les maladies rares. L’objectif de mon travail était d'explorer la faisabilité, les contraintes et l'apport de l'approche Bayésienne dans les essais cliniques portant sur des cancers rares lorsque le critère principal est censuré. Tout d’abord, une revue de la littérature a confirmé la faible implémentation actuelle des méthodes Bayésiennes dans l'analyse des essais cliniques avec critère de survie.Le second axe de ce travail a porté sur le développement d’un essai Bayésien avec critère de survie, intégrant des données historiques, dans le cadre d’un essai réel portant sur une pathologie rare (ostéosarcome). Le prior intégrait des données historiques individuelles sur le bras contrôle et des données agrégées sur l’effet relatif du traitement. Une large étude de simulations a permis d’évaluer les caractéristiques opératoires du design proposé, de calibrer le modèle, tout en explorant la problématique de la commensurabilité entre les données historiques et actuelles. Enfin, la ré-analyse de trois essais cliniques publiés a permis d’illustrer l'apport de l'approche Bayésienne dans l'expression des résultats et la manière dont cette approche permet d’enrichir l’analyse fréquentiste d’un essai. / Bayesian approach augments the information provided by the trial itself by incorporating external information into the trial analysis. In addition, this approach allows the results to be expressed in terms of probability of some treatment effect, which is more informative and interpretable than a p-value and a confidence interval. In addition, the frequent reduction of an analysis to a binary interpretation of the results (significant versus non-significant) is particularly harmful in rare diseases.In this context, the objective of my work was to explore the feasibility, constraints and contribution of the Bayesian approach in clinical trials in rare cancers with a primary censored endpoint. A review of the literature confirmed that the implementation of Bayesian methods is still limited in the analysis of clinical trials with a censored endpoint.In the second part of our work, we developed a Bayesian design, integrating historical data in the setting of a real clinical trial with a survival endpoint in a rare disease (osteosarcoma). The prior incorporated individual historical data on the control arm and aggregate historical data on the relative treatment effect. Through a large simulation study, we evaluated the operating characteristics of the proposed design and calibrated the model while exploring the issue of commensurability between historical and current data. Finally, the re-analysis of three clinical trials allowed us to illustrate the contribution of Bayesian approach to the expression of the results, and how this approach enriches the frequentist analysis of a trial.

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.

Made available in DSpace on 2016-06-02T20:06:02Z (GMT). No. of bitstreams: 1 2203.pdf: 1300161 bytes, checksum: 2c1f11d939eab9ab849bb04bf2363a53 (MD5) Previous issue date: 2009-02-10 / Financiadora de Estudos e Projetos / This study considers two discrete distributions based on Bernoulli trials: the Binomial and the Negative Binomial. We explore credibility and confidence intervals to estimate the probability of success of each distribution. The main goal is to analyze their performance coverage probability and average range across the parametric space. We also consider point analysis of bayesian estimators and maximum likelihood estimators, whose interest is to confirm through simulation their consistency, bias and mean square error. In this paper the Objective Bayesian Inference is applied through the noninformative Bayes-Laplace prior, Haldane prior, reference prior and least favorable prior. By analyzing the prior distributions in the minimax decision theory context we verified that the least favorable prior distribution has every other considered prior distributions as particular cases when a quadratic loss function is applied, and matches the Bayes-Laplace prior in considering the quadratic weighed loss function for the Binomial model (which was never found in literature). We used the noninformative Bayes-Laplace prior and Jeffreys prior for the Negative Binomial model. Our findings show through coverage probability, average range of bayesian intervals and point estimation that the Objective Bayesian Inference has good frequentist properties for the probability of success of Binomial and Negative Binomial models. The last stage of this study discusses the presence of correlated proportions in matched-pairs (2 × 2 table) of Bernoulli with the goal of obtaining more information in relation of the considered measures for testing the occurrence of correlated proportions. In this sense the Trinomial model and the partial likelihood function were used from the frequentist and bayesian point of view. The Full Bayesian Significance Test (FBST) was used for real data sets and was shown sensitive to parameterization, however, this study was not possible for the frequentist method since distinct methods are needed to be applied to Trinomial model and the partial likelihood function. / Neste estudo são abordadas duas distribuições discretas baseadas em ensaios de Bernoulli, a Binomial e a Binomial Negativa. São explorados intervalos de credibilidade e confiança para estimação da probabilidade de sucesso de ambas as distribuições. A principal finalidade é analisar nos contextos clássico e bayesiano o desempenho da probabilidade de cobertura e amplitude média gerada pelos intervalos de confiança e intervalos de credibilidade ao longo do espaço paramétrico. Considerou-se também a análise dos estimadores pontuais bayesianos e o estimador de máxima verossimilhança, cujo interesse é confirmar por meio de simulação a consistência e calcular o viés e o erro quadrático médio dos mesmos. A Inferência Bayesiana Objetiva é empregada neste estudo por meio das distribuições a priori não-informativas de Bayes-Laplace, de Haldane, de Jeffreys e menos favorável. Ao analisar as distribuições a priori no contexto de teoria de decisões minimax, a distribuição a priori menos favorável resgata as demais citadas ao empregar a função de perda quadrática e coincide com a distribuição a priori de Bayes-Laplace ao considerar a função de perda quadrática ponderada para o modelo Binomial, o que não foi encontrado até o momento na literatura. Para o modelo Binomial Negativa são consideradas as distribuições a priori não-informativas de Bayes-Laplace e de Jeffreys. Com os estudos desenvolvidos pôde-se observar que a Inferência Bayesiana Objetiva para a probabilidade de sucesso dos modelos Binomial e Binomial Negativa apresentou boas propriedades freqüentistas, analisadas a partir da probabilidade de cobertura e amplitude média dos intervalos bayesianos e por meio das propriedades dos estimadores pontuais. A última etapa do trabalho consiste na análise da ocorrência de proporções correlacionadas em pares de eventos de Bernoulli (tabela 2×2) com a finalidade de determinar um possível ganho de informação em relação as medidas consideradas para testar a ocorrência de proporções correlacionadas. Para tanto fez-se uso do modelo Trinomial e da função de verossimilhança parcial tanto numa abordagem clássica quanto bayesiana. Nos conjuntos de dados analisados observou-se a medida de evidência bayesiana (FBST) como sensível à parametrização, já para os métodos clássicos essa comparação não foi possível, pois métodos distintos precisam ser aplicados para o modelo Trinomial e para a função de verossimilhança parcial.