Frequentemente em pesquisas médicas ou epidemiológicas, múltiplas medidas de um mesmo sujeito são tomadas ao longo do tempo, caracterizando um estudo com dados longitudinais. Nos últimos anos, a técnica estatística que tem sido mais utilizada para a análise desses estudos é o modelo misto, pois este permite incorporar a provável correlação das observações de um mesmo indivíduo e é flexível para lidar com situações de desbalanceamento e dados faltantes. Um modelo misto muito utilizado nesses casos é o modelo de coeficientes aleatórios, que permite que a relação da variável resposta com o tempo seja descrita através de uma função matemática. Há duas abordagens para se analisar o modelo misto: clássica e Bayesiana. Para se aplicar o método Bayesiano nesse caso, deve-se utilizar um dos métodos MCMC, já que a distribuição a posteriori não é analiticamente derivável. O método MCMC utilizado neste trabalho foi o Amostrador de Gibbs. As duas abordagens podem levar a resultados diferentes, deixando o pesquisador em dúvida sobre qual método utilizar. Poucos estudos compararam a abordagem clássica e a Bayesiana para a análise de dados longitudinais via modelos mistos. O objetivo deste trabalho é proceder a um estudo de simulação, para comparar estas abordagens em termos de vício e precisão. Compararam-se os estimadores Bayesianos média, moda (estimada não-parametricamente) e mediana a posteriori e o estimador clássico obtido pelo método REML. Simulou-se um modelo que envolve efeito de tempo e de uma covariável denominada tratamento, assumindo a estrutura Componentes de Variância para as duas matrizes de covariâncias do modelo misto. Diferentes configurações de tamanho de amostra e desbalanceamento foram adotadas, para avaliar o desempenho dos métodos frente a essas situações. Em cada uma das configurações, foram realizadas 1000 replicações no software R. As maiores diferenças encontradas foram em relação a alguns componentes de variância, sendo que pelo menos um dos estimadores Bayesianos apresentou erro quadrático médio menor do que o estimador clássico em todas as configurações. Não foi possível identificar um único estimador Bayesiano como sendo o melhor para todos os casos estudados. A moda mostrou-se um estimador com boas propriedades em algumas situações e por isso sugere-se que o mesmo seja implementado no software WinBUGS. / Frequently in medical or epidemiologic research, multiple measures of the same subject are observed over the time. This characterizes a study with longitudinal data. In the last years, the statistical technique that has been used in the analysis of these studies is the mixed model, because it permits to model the probable correlation between the observations of the same individual, and it handles well with situations of unbalanceament and missing data. A mixed model that has been used in these cases is the random coefficients model. It allows the relationship between the response variable and the time to be described through a mathematical function. There are two approaches to analyze mixed models: the classical and the Bayesian methods. In order to apply the Bayesian inference in this case, the MCMC methods have to be used, because the posterior distribution is not analytically derivable. The MCMC method used in this work was the Gibbs Sampler. The two approaches may produce different results, and because of that the researcher may be in doubt of what method to use. In the literature, there are few studies that compare the classical and the Bayesian approach in the analysis of longitudinal data using mixed models with an analytical way. So, the objective of this work is to proceed a simulation study to compare the Bayesian and classical inferences in terms of bias and precision. The Bayesian estimators posterior mean, (non-parametric) mode and median, and the classical estimator obtained by the REML method were compared. We simulated a model that involves the effects of time and a covariable named “treatment”, and that assumes a Variance Components structure for the two covariance matrices of the mixed model. Different configurations of sample size and unbalanced data were adopted, to see the performance of the methods with these situations. In each of these configurations, 1000 replications were proceeded in the R software. The main differences founded were in the estimation of the variance components of the random effects. For these parameters, one of the Bayesian estimators had lower mean square error than the classical in all of the configurations. It was not possible to identify one best Bayesian estimator in all the cases studied. The mode seems an estimator with good properties in some situations and because of that it should be implemented in the WinBUGS software.

Submitted by MARCOS LEANDRO TEIXEIRA DE OLIVEIRA (marcosteixeira@ufv.br) on 2018-07-30T13:38:55Z No. of bitstreams: 1 texto completo.pdf: 2369640 bytes, checksum: a88843a731f1549741b5651418bd3968 (MD5) / Made available in DSpace on 2018-07-30T13:38:55Z (GMT). No. of bitstreams: 1 texto completo.pdf: 2369640 bytes, checksum: a88843a731f1549741b5651418bd3968 (MD5) Previous issue date: 2018-02-22 / O Teorema de Bayes descreve a probabilidade de um evento em termos de informações a priori, comumente estudada como Probabilidade Condicional. Porém, essa relação permite que a probabilidade seja interpretada como uma medida da plausibilidade de uma expectativa ou de uma crença pessoal. Dessa forma, a Probabilidade Bayesiana pode ser usada para validar estatisticamente se uma teoria científica está correta frente aos resultados experimentais ou se um acusado ́e culpado frente às provas apresentadas. Nesse projeto, será explorado o conceito de Probabilidade Bayesiana, apresentado sua fundamentação teórica e serão propostas formas de trabalhar o uso da lógica, probabilidade e estatística com os alunos do ensino médio. Serão propostas sugestões de aplicação do Teorema de Bayes em sala de aula de forma contextualizada em duas frentes: Utilizando a Probabilidade Condicional e a linguagem Inferência de Parâmetros. Em ambos os casos as sugestões conterão dicas para a aplicação do conteúdo no ensino médio, como introdução atividade, desenvolvimento, questionamentos, suporte pedagógico, além das resoluções com detalhes / The Bayes Theorem describes the probability of an event based on some a priori information, this concept is usually studied as the conditional probability. This relation allows the probability to be interpreted as a plausibility measure of a person expectations and believes. Therefore, the Bayesian Probability can be employed to statistically validate a scientific theory face the experimental results or if a accused is guilt taking in account the evidence. This work will explore the Bayesian Probability concept presenting its theory and also will propose ways to work the concepts of Logic, Probability and Statistics with high school students. Classroom activities will be presented to explore contextualized applications of the Theorem of Bayes. Two approaches will be presented, the first using only the Conditional Probability formula and a second showing how to make some simple parameter inferences. On each case we will present suggestions, hints and a detailed step by step to the teacher that desires to develop these contend with their high school students

This diploma thesis is about Bayesian approach in managerial decision making process. The goal is not only to quantify the principals used by managers during decision making in real situations but also the application of Bayesian methods on given examples. The mentioned principals are the probability updates after gaining a new information. In thesis are also described the computer systems which work with Bayesian calculations and a chosen system is described in detail. In a practical part of this thesis is demonstrated the use of Bayesian principals in real decision making situations -- there is demonstrated the use of Bayesian games, Bayesian networks (both classic and dynamic) and risk decision making process.

Purpose: To investigate reasons for patients not proceeding to resective epilepsy surgery after subdural grid evaluation (SDE). To correlate noninvasive investigation results with invasive EEG observations in a set of patients with nonlesional brain MRIs. Methods: Retrospective study of adult epilepsy patients undergoing SDE during an 8-year period at Cleveland Clinic. Construction of semiquantitative "scores" and Bayesian predictors summarizing the localizing value and concordance between noninvasive parameters in a subset with nonlesional MRIs. Results: One hundred forty patients underwent SDE, 25 of whom were subsequently denied resective surgery. In 10 of 25, this was caused by a nonlocalizing subdural ictal EEG onset. Eight of 10 such patients were nonlesional on MRI. Among all nonlesional patients (n = 34 of 140), n 1 = 10 of 34 patients had nonlocalizing and n2 = 24 of 34 had localizing, subdural ictal onsets. As groups, n1 and n 2 were statistically disjoint relative to their noninvasive scores. Bayesian measures predictive of focal invasive ictal EEG were highest for complete concordance of noninvasive parameters, decreasing with lesser degrees of concordance. A localizing scalp interictal EEG was a particularly good Bayesian prognosticator. Conclusions: A small but significant proportion of SDE patients are denied subsequent therapeutic resective surgery. This is due to several reasons, including a nonlocalizing intracranial ictal EEG. The majority of such patients have nonlesional MRIs. The noninvasive data may be summarized by a semiquantitative score, as well as Bayesian likelihood ratios, which correlate with subsequent invasive outcome. This approach may find use in the selection and counseling of potential surgical candidates offered SDE.

Understanding human navigation behavior has implications for a wide range of application scenarios. For example, insights into geo-spatial navigation in urban areas can impact city planning or public transport. Similarly, knowledge about navigation on the web can help to improve web site structures or service experience. In this work, we focus on a hypothesis-driven approach to address the task of understanding human navigation: We aim to formulate and compare ideas — for example stemming from existing theory, literature, intuition, or previous experiments — based on a given set of navigational observations. For example, we may compare whether tourists exploring a city walk “short distances” before taking their next photo vs. they tend to "travel long distances between points of interest", or whether users browsing Wikipedia "navigate semantically" vs. "click randomly". For this, the Bayesian method HypTrails has recently been proposed. However, while HypTrails is a straightforward and flexible approach, several major challenges remain: i) HypTrails does not account for heterogeneity (e.g., incorporating differently behaving user groups such as tourists and locals is not possible), ii) HypTrails does not support the user in conceiving novel hypotheses when confronted with a large set of possibly relevant background information or influence factors, e.g., points of interest, popularity of locations, time of the day, or user properties, and finally iii) formulating hypotheses can be technically challenging depending on the application scenario (e.g., due to continuous observations or temporal constraints). In this thesis, we address these limitations by introducing various novel methods and tools and explore a wide range of case studies. In particular, our main contributions are the methods MixedTrails and SubTrails which specifically address the first two limitations: MixedTrails is an approach for hypothesis comparison that extends the previously proposed HypTrails method to allow formulating and comparing heterogeneous hypotheses (e.g., incorporating differently behaving user groups). SubTrails is a method that supports hypothesis conception by automatically discovering interpretable subgroups with exceptional navigation behavior. In addition, our methodological contributions also include several tools consisting of a distributed implementation of HypTrails, a web application for visualizing geo-spatial human navigation in the context of background information, as well as a system for collecting, analyzing, and visualizing mobile participatory sensing data. Furthermore, we conduct case studies in many application domains, which encompass — among others — geo-spatial navigation based on photos from the photo-sharing platform Flickr, browsing behavior on the social tagging system BibSonomy, and task choosing behavior on a commercial crowdsourcing platform. In the process, we develop approaches to cope with application specific subtleties (like continuous observations and temporal constraints). The corresponding studies illustrate the variety of domains and facets in which navigation behavior can be studied and, thus, showcase the expressiveness, applicability, and flexibility of our methods. Using these methods, we present new aspects of navigational phenomena which ultimately help to better understand the multi-faceted characteristics of human navigation behavior. / Menschliches Navigationsverhalten zu verstehen, kann in einer Reihe von Anwendungsgebieten große Fortschritte bringen. Zum Beispiel können Einblicke in räumliche Navigation, wie etwa in Innenstädten, dabei helfen Infrastrukturen und öffentliche Verkehrsmittel besser abzustimmen. Genauso kann Wissen über das Navigationsverhalten von Benutzern im Internet, Entwickler dabei unterstützen Webseiten besser zu strukturieren oder generell die Benutzererfahrung zu verbessern. In dieser Arbeit konzentrieren wir uns auf einen Hypothesen-getriebenen Ansatz, um menschliches Navigationsverhalten zu verstehen. Das heißt, wir formulieren und vergleichen Hypothesen basierend auf beobachteten Navigationspfaden. Diese Hypothesen basieren zumeist auf existierenden Theorien, Literatur, vorherigen Experimenten oder Intuition. Beispielsweise kann es interessant sein zu vergleichen, ob Touristen, die eine Stadt erkunden, eher zu nahegelegenen Sehenswürdigkeiten laufen, als vornehmlich große Strecken zurückzulegen. Weiterhin kann man in Online-Szenarien vergleichen, ob Benutzer zum Beispiel auf Wikipedia eher semantisch navigieren, als zufällig Artikel anzusurfen. Für diese Szenarien wurde HypTrails entwickelt, ein Bayes’scher Ansatz zum Vergleich von Navigationshypothesen. Doch obwohl HypTrails eine einfach zu benutzende und sehr flexible Methode darstellt, hat es einige deutliche Schwachstellen: Zum einen kann HypTrails keine heterogenen Prozesse modellieren (z.B., um das Verhalten von ver- schiedenen Nutzergruppen, wie etwa von Touristen und Einheimischen, zu unterscheiden). Außerdem bietet HypTrails dem Benutzer keine Unterstützung bei der Entwicklung neuer Hypothesen. Dies stellt vor allem in Kombination mit großen Mengen an Hintergrundinformationen und anderen Einflussgrößen (z.B., Sehenswürdigkeiten, Beliebtheit von Orten, Tageszeiten, oder verschieden Benutzereigenschaften) eine große Herausforderung dar. Außerdem kann sich das Formulieren von adäquaten Hypothesen abhängig vom Anwendungsszenario als schwierig erweisen (z.B. aufgrund von kontinuierlichen, räumlichen Koordinaten oder zeitlichen Nebenbedingungen). In dieser Arbeit setzen wir an eben jenen Problemstellungen an. Unsere Hauptbeiträge bestehen dabei aus den Ansätzen MixedTrails und SubTrails, die vor allem die ersten beiden genannten Schwachstellen adressieren: MixedTrails stellt einen Ansatz zum Vergleich von Hypothesen dar, der auf HypTrails basiert, es aber ermöglicht heterogene Hypothesen zu formulieren und zu vergleichen (z.B., bei Benutzergruppen mit unterschiedlichem Bewegungsverhalten). Während SubTrails eine Methode darstellt, die das Entwickeln neuer Hypothesen unterstützt, indem es die automatische Entdeckung von interpretierbaren Subgruppen mit außergewöhnlichen Bewegungscharakteristiken ermöglicht. Weiterhin, stellen wir eine verteitle und hochparallele Implementierung von HypTrails, ein Werkzeug zur Visualisierung von räumlicher Navigation zusammen mit Hintergrundinformationen, sowie ein System zur Sammlung, Analyse und Visualisierung von Daten aus dem Bereich des Participatory Sensing vor. Schließlich, führen wir mehrere Studien in verschiedenen Anwendungsbereichen durch. Wir untersuchen etwa räumliche Navigation basierend auf Photos der Onlineplattform Flickr, Browsing-Verhalten der Nutzer auf dem Verschlagwortungssystem BibSonomy, und das Arbeitsverhalten von Nutzern einer kommerziellen Crowdsourcing-Plattform. Dabei entwickeln wir mehrere Ansätze, um mit den Eigenheiten der spezifischen Szenarien umgehen zu können (wie etwa kontinuierliche räumliche Koordinaten oder zeitliche Nebenbedingungen). Die Ergebnisse zeigen die Vielzahl von Anwendungsgebieten und Facetten, in denen Navigationsverhalten analysiert werden kann, und illustrieren so die Ausdrucksstärke, vielseitige Anwendbarkeit und Flexibilität unserer Methoden. Gleichzeitig, geben wir neue Einblicke in verschiedene Navigationsprozesse und ermöglichen so einen wichtigen Schritt hin zum Verständnis der vielfältigen Ebenen menschlichen Navigationsverhaltens.

<p> Bayes' formula may be written as follows: </p> <p> P(yᵢ|X) = P(X|yᵢ)・P(yᵢ)/j=K Σ j=1 P(X|yⱼ)・P(yⱼ) where (1) </p> <p> Y = {y₁, y₂,..., y_K} </p> <P> X = {x₁, x₂,..., xₖ} </p> <p> Assuming independence of attributes x₁, x₂,..., xₖ, Bayes' formula may be rewritten as follows: </p> <p> P(yᵢ|X) = P(x₁|yᵢ)・P(x₂|yᵢ)・...・P(xₖ|yᵢ)・P(yᵢ)/j=K Σ j=1 P(x₁|yⱼ)・P(x₂|yⱼ)・...・P(xₖ|yⱼ)・P(yⱼ) (2) </p> <p> In medical diagnosis the y's denote disease states and the x's denote the presence or absence of symptoms. Bayesian inference is applied to medical diagnosis as follows: for an individual with data set X, the predicted diagnosis is the disease yⱼ such that P(yⱼ|X) = max_i P(yᵢ|X), i=1,2,...,K (3) </p> <p> as calculated from (2). </p> <p> Inferences based on (2) and (3) correctly allocate a high proportion of patients (>70%) in studies to date, despite violations of the independence assumption. The aim of this thesis is modest, (i) to demonstrate the applicability of Bayesian inference to the problem of medical diagnosis (ii) to review pertinent literature (iii) to present a Monte Carlo method which simulates the application of Bayes' formula to distinguish among diseases (iv) to present and discuss the results of Monte Carlo experiments which allow statistical statements to be made concerning the accuracy of Bayesian inference when the assumption of independence is violated. </p> <p> The Monte Carlo study considers paired dependence among attributes when Bayes' formula is used to predict diagnoses from among 6 disease categories. A parameter which measured deviations from attribute independence is defined by DH=(j=6 Σ j=1|P(x_B|x_A,yⱼ)-P(x_B|yⱼ)|)/6, where x_A and x_B denote a dependent attribute pair. It was found that the correct number of Bayesian predictions, M, decreases markedly as attributes increasing diverge from independence, ie, as DH increases. However, a simple first order linear model of the form M = B₀+B₁・DH does not consistently explain the variation in M. </p> / Thesis / Master of Science (MSc)

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