In most eukaryotes, alternative splicing is an important regulatory mechanism of gene expression that results in a single gene coding for multiple protein isoforms, thus largely increases the diversity of the proteome. RNA-seq is widely used for genome-wide splicing isoform quantification, and several effective and powerful methods have been developed for splicing analysis with RNA-seq data. However, it remains problematic for genes with low coverages or large number of isoforms. These difficulties may in principle be ameliorated by exploiting correlations encoded in the structured data sources. This thesis contributes to developments of Bayesian methods for splicing analysis by leveraging additional information in multiple datasets with structured prior distributions. First, we developed DICEseq, the first isoform quantification method tailored to time-series RNA-seq experiments. DICEseq explicitly models the correlations between experiments at different time points to aid the quantification of isoforms across experiments. Numerical experiments on both simulated and real datasets show that DICEseq yields more accurate results than state-of-the-art methods, an advantage that can become considerable at low coverage levels. Furthermore, DICEseq permits to quantify the trade-off between temporal sampling of RNA and depth of sequencing, frequently an important choice when planning experiments. Second, we developed BRIE (Bayesian Regression for Isoform Estimation), a Bayesian hierarchical model which resolves the difficulties in splicing analysis in single-cell RNA-seq (scRNA-seq) data by learning an informative prior distribution from sequence features. This method combines the quantification and imputation for splicing analysis via a Bayesian way, which is particularly useful in scRNA-seq data due to its extreme low coverages and high technical noises. We validated BRIE on several scRNA-seq data sets, showing that BRIE yields reproducible estimates of exon inclusion ratios in single cells. Third, we provided an effective tool by using Bayes factor to sensitively detect differential splicing between different single cells. When applying BRIE to a few real datasets, we found interesting heterogeneity patterns in splicing events across cell population, for example alternative exons in DNMT3B. In summary, this thesis proposes structured Bayesian methods to integrate multiple datasets to improve splicing analysis and study its biological functions.

This dissertation consists of three chapters on semi-parametric Bayesian Econometric methods. Chapter 1 applies a semi-parametric method to demand systems, and compares the abilities to recover the true elasticities of different approaches to linearly estimating the widely used Almost Ideal demand model, by either iteration or approximation. Chapter 2 co-authored with Dr. Melvyn Weeks introduces a new semi-parametric Bayesian Generalized Least Square estimator, which employs the Dirichlet Process prior to cope with potential heterogeneity in the error distributions. Two methods are discussed as special cases of the GLS estimator, the Seemingly Unrelated Regression for equation systems, and the Random Effects Model for panel data, which can be applied to many fields such as the demand analysis in Chapter 1. Chapter 3 focuses on the subset selection for the efficiencies of firms, which addresses the influence of heterogeneity in the distributions of efficiencies on subset selections by applying the semi-parametric Bayesian Random Effects Model introduced in Chapter 2.

Este trabalho objetiva investigar os padrões espaço-temporais dos casos notificados de AIDS entre os anos de 2000 a 2010 no município de São Paulo, SP, segundo seus 96 Distritos Administrativos, bem como suas associações com as características sociodemográficas e de vulnerabilidade social destas áreas. Trata-se de um estudo ecológico, e as ferramentas utilizadas para a estimação de taxas de incidência e análise dos dados são modelos bayesianos de regressão que incorporam efeitos temporais e espaciais. Estes modelos incluem efeitos aleatórios com uma distribuição CAR (autorregressiva condicional) normal bivariada que capturam a influência das áreas adjacentes sobre o número de casos notificados em cada região, segundo o sexo. Estimativas dos parâmetros do modelo foram obtidas pelo método de simulação estocástica MCMC (Monte Carlo em Cadeia de Markov). O padrão espaço-temporal encontrado neste trabalho mostra traços históricos da epidemia de AIDS no Brasil e o estudo estratificado por sexo na evolução da doença para os anos em estudo evidencia aspectos comportamentais da transmissão em populações de risco. Pode-se evidenciar que os Distritos Administrativos com classes econômicas melhores apresentaram maiores incidência da doença no sexo masculino, já áreas com classes econômicas menos favorecidas apresentaram maiores taxas de incidência da AIDS no sexo feminino, justificado pela pauperização e feminização da AIDS. Os resultados encontrados são úteis para subsidiar o planejamento de políticas e ações de saúde direcionadas para o controle da AIDS em áreas e populações de risco diversificada por sexo do município de São Paulo. Ressalta-se a necessidade de atuação diversificada por Distrito Administrativo no acesso a serviços de saúde, entre os sexos, com o intuito de propor uma estratégia de atendimento para homens e mulheres separadas por áreas e provendo equipes de apoio diferentes para área com características particulares. A abordagem bayesiana proposta se mostrou satisfatório em indicar áreas que precisam de mais atenção quanto a necessidade de serviços de saúde, desenvolvimento humano e vulnerabilidade. / This study aims to investigate the spatiotemporal patterns of reported AIDS cases between the years 2000 to 2010 in São Paulo, SP, according to its 96 Administrative Districts, and their associations with sociodemographic characteristics and social vulnerability of these areas. This is an ecological study, and the tools used to incidence rates estimation and analysis of data is Bayesian regression models that incorporate spatial and temporal effects. These models include random effects with a CAR distribution (conditional autoregressive) bivariate normal that capture the influence of adjacent fields on the number of cases reported in each region, according to sex. Estimates of the model parameters were obtained by stochastic simulation method MCMC (Monte Carlo Markov Chain). The spatiotemporal pattern found in this work shows historical traces of the AIDS epidemic in Brazil and the study stratified by gender in the course of the disease for the years under study reveals behavioral aspects of transmission in populations at risk. Can show that the Administrative Districts with better economic classes had higher incidence of the disease in males, since areas with less favored economic classes had higher incidence rates of AIDS among women, justified by the AIDS pauperization and feminization. The results are useful to support the policy planning and health actions directed to the control of AIDS in areas and populations of diverse risk by gender in São Paulo. It emphasizes the need for diversified operations by Administrative District in access to health services, gender, in order to propose a service strategy to separate men and women for areas and providing different support teams to the area with particular characteristics. The bayesian approach proposed proved satisfactory to indicate areas that need more attention as the need for health services, human development and vulnerability.

Este trabalho objetiva investigar os padrões espaço-temporais dos casos notificados de AIDS entre os anos de 2000 a 2010 no município de São Paulo, SP, segundo seus 96 Distritos Administrativos, bem como suas associações com as características sociodemográficas e de vulnerabilidade social destas áreas. Trata-se de um estudo ecológico, e as ferramentas utilizadas para a estimação de taxas de incidência e análise dos dados são modelos bayesianos de regressão que incorporam efeitos temporais e espaciais. Estes modelos incluem efeitos aleatórios com uma distribuição CAR (autorregressiva condicional) normal bivariada que capturam a influência das áreas adjacentes sobre o número de casos notificados em cada região, segundo o sexo. Estimativas dos parâmetros do modelo foram obtidas pelo método de simulação estocástica MCMC (Monte Carlo em Cadeia de Markov). O padrão espaço-temporal encontrado neste trabalho mostra traços históricos da epidemia de AIDS no Brasil e o estudo estratificado por sexo na evolução da doença para os anos em estudo evidencia aspectos comportamentais da transmissão em populações de risco. Pode-se evidenciar que os Distritos Administrativos com classes econômicas melhores apresentaram maiores incidência da doença no sexo masculino, já áreas com classes econômicas menos favorecidas apresentaram maiores taxas de incidência da AIDS no sexo feminino, justificado pela pauperização e feminização da AIDS. Os resultados encontrados são úteis para subsidiar o planejamento de políticas e ações de saúde direcionadas para o controle da AIDS em áreas e populações de risco diversificada por sexo do município de São Paulo. Ressalta-se a necessidade de atuação diversificada por Distrito Administrativo no acesso a serviços de saúde, entre os sexos, com o intuito de propor uma estratégia de atendimento para homens e mulheres separadas por áreas e provendo equipes de apoio diferentes para área com características particulares. A abordagem bayesiana proposta se mostrou satisfatório em indicar áreas que precisam de mais atenção quanto a necessidade de serviços de saúde, desenvolvimento humano e vulnerabilidade. / This study aims to investigate the spatiotemporal patterns of reported AIDS cases between the years 2000 to 2010 in São Paulo, SP, according to its 96 Administrative Districts, and their associations with sociodemographic characteristics and social vulnerability of these areas. This is an ecological study, and the tools used to incidence rates estimation and analysis of data is Bayesian regression models that incorporate spatial and temporal effects. These models include random effects with a CAR distribution (conditional autoregressive) bivariate normal that capture the influence of adjacent fields on the number of cases reported in each region, according to sex. Estimates of the model parameters were obtained by stochastic simulation method MCMC (Monte Carlo Markov Chain). The spatiotemporal pattern found in this work shows historical traces of the AIDS epidemic in Brazil and the study stratified by gender in the course of the disease for the years under study reveals behavioral aspects of transmission in populations at risk. Can show that the Administrative Districts with better economic classes had higher incidence of the disease in males, since areas with less favored economic classes had higher incidence rates of AIDS among women, justified by the AIDS pauperization and feminization. The results are useful to support the policy planning and health actions directed to the control of AIDS in areas and populations of diverse risk by gender in São Paulo. It emphasizes the need for diversified operations by Administrative District in access to health services, gender, in order to propose a service strategy to separate men and women for areas and providing different support teams to the area with particular characteristics. The bayesian approach proposed proved satisfactory to indicate areas that need more attention as the need for health services, human development and vulnerability.

Dados de s´eries temporais s~ao originados a partir de estudos em que se reportam, por exemplo, taxas de mortalidade, n´umero de hospitaliza¸c~oes, de infec¸c~oes por alguma doen¸ca ou outro evento de interesse, em per´?odos definidos (dia, semana, m^es ou ano), objetivando-se observar tend^encias, sazonalidades ou fatores associados. Dados de contagem s~ao aqueles representados pelas vari´aveis quantitativas discretas, ou seja, observa¸c~oes que assumem valores inteiros, no intervalo {0, 1, 2, 3, ...}, por exemplo, o n´umero de filhos de casais residentes em um bairro. Diante dessa particularidade, ferramentas estat´?sticas adequadas devem ser utilizadas, e modelos baseados na distribui¸c~ao de Poisson apresentam-se como op¸c~oes mais indicadas do que os baseados nos m´etodos propostos por Box e Jenkins (2008), usualmente utilizados para an´alise de dados cont´?nuos, mas empregados para dados discretos, ap´os transforma¸c~oes logar´?tmicas. Uma limita¸c~ao da distribui¸c~ao de Poisson ´e que ela assume m´edia e vari^ancia iguais, sendo um obst´aculo nos casos em que h´a superdispers~ao (vari^ancia maior que a m´edia) ou subdispers~ao (vari^ancia menor que a m´edia). Diante disso, a distribui¸c~ao Poisson Dupla, proposta por Efron (1986), surge como alternativa, pois permite se estimarem os par^ametros de m´edia e vari^ancia, nos casos em que a vari^ancia dos dados ´e menor, igual ou maior que a m´edia, fornecendo grande flexibilidade aos modelos. Este trabalho teve como objetivo principal o desenvolvimento de modelos Bayesianos de s´eries temporais para dados de contagem, utilizando-se distribui¸c~oes de probabilidade para vari´aveis discretas, tais como de Poisson e Poisson Dupla. Al´em disso, foi introduzido um modelo baseado na distribui¸c~ao Poisson Dupla para dados de contagem com excesso de zeros. Os resultados obtidos pelo ajuste dos modelos de s´eries temporais baseados na distribui¸c~ao Poisson Dupla foram comparados com aqueles obtidos por meio do uso da distribui¸c~ao de Poisson. Como aplica¸c~oes principais, foram apresentados resultados obtidos pelo ajuste de modelos para dados de registros de acidentes com picadas de cobras, no Estado de S~ao Paulo, e picadas de escorpi~oes, na cidade de Ribeir~ao Preto, SP, entre os anos de 2007 e 2014. Com rela¸c~ao a esta ´ultima aplica¸c~ao, foram consideradas covari´aveis referentes a dados clim´aticos, como temperaturas m´aximas e m´?nimas m´edias mensais e precipita¸c~ao. Nas situa¸c~oes em que a vari^ancia era diferente da m´edia, modelos baseados na distribui¸c~ao Poisson Dupla mostraram melhor ajuste aos dados, quando comparados aos modelos de Poisson. / Time series data are derived from studies in which there are reported mortality, number of hospitalizations infections by disease or other event of interest per day, week, month or year, in order to observe trends, seasonality or associated factors. Count data are represented by discrete quantitative variables, i.e. observations that take integer values in the range {0, 1, 2, 3, ...}. In view of this particular characteristic, such data must be analyzed by adequate statistical tools and the Poisson distribution is an option for modeling, being more suitable than models based on methods proposed by Box and Jenkins (2008), usually applied for continuous data, but used in the modeling of discrete data after logarithmic transformation. A limitation of the Poisson distribution is that it assumes equal mean and variance being an obstacle in cases which there are data overdispersion (variance higher than mean) or underdispersion (variance lower than mean). Therefore the Double Poisson distribution, proposed by Efron (1986), is an alternative because it allows to estimate the mean and variance parameters in cases wich variance of the data is lower, equal, or higher than mean providing great flexibility to the models. This work aims to develop time series models for count data, under Bayesian approach using probability distributions for discrete variables such as Poisson and Double Poisson. Furthermore it will be introduced a zero-inflated Double Poisson model to excess zeros counting data. The results obtained by adjusting the time series models based on Double Poisson distribution are compared with those obtained by considering the Poisson distribution. As main applications modeling of snake bites reports in the State of S~ao Paulo and scorpion stings in the city of Ribeir~ao Preto considering covariates as maximum and minimum average monthly temperatures and rainfall among the years 2007 and 2014 will be presented. Regression models based on double Poisson distribution showed a better fit to the data, when compared to Poisson models.

O estudo de métodos estatísticos na avaliação de métodos diagnósticos tem aumentado consideravelmente nas últimas décadas. Desde o início, quando Yerushalmy (1947) publicou seu traba lho sobre confiabilidade do roentgeno grama na identificação da tuberculose, novas metodologias surgiram para que fosse possível a obtenção de valores de sensibilidade e especificidade de testes diagnósticos. A sensibilidade é definida como a probabilidade de o teste sob investigação fornecer um resultado positivo, dado que o indivíduo é realmen te portador da enfermidade. A especifi cidade, por sua vez, é definida como a probabilidade de o teste fornecer um resultado negativo, dado que o indivíduo está livre da enfermidade. Na prática, é comum ocorrerem situações em que uma proporção de indivíduos selecionados não pode ter o estado real da doença verificado, por se tratar de procedimentos invasivos, como no diagnóstico de câncer de pulmão, ou quaisquer outros casos em que são envolvidos riscos, portanto não praticá veis, nem éticos, ou ainda por serem de alto custo. Assim, em vez de se contornar o proble ma, muitos estudos de avaliação de performance de testes diagnósticos são elaborados apenas com informações de indivíduos verificados. Esse procedimento pode levar a resultados viesados. É o chamado viés de verificação, que consiste no cálculo de estimativas de sensibilidade e especi ficidade de testes diagnósticos quando apenas os indivíduos verificados pelo padrão ouro são inseridos na análise e os não verificados são descartados ou considerados livres de doença. Este trabalho apresenta uma revisão das metodologias já propostas para se calcularem a sensibilidade e a especificidade quando existe o viés de verificação, bem como uma análise detalhada da influência da proporção de indivíduos não verificados, o efeito do tamanho amostral e a escolha de distri buições a priori, quando utilizada a metodologia bayesiana, no cálculo dessas estimativas. Também foi introduzida uma metodologia, sob enfoque bayesiano, para a estimação das medidas de desempenho de dois testes diagnósticos, na presen ça do viés de verificação. / The study of statistical methods on diagnostic tests evaluation has increa sed in the last decades. Since the beginning, when Yerushalmy (1947) published his work about trustwor thiness of the roentgenogram in the identification of the tuberculosis, new methodologies had appeared and so that it was possible to get values of sensi tivity and specificity of diagnostic tests. Sensitivity is defined as the probability of the test under inquiry supply a positive result, since that the individual is really carrying on the disease. The specificity, in the other hand, is defined as the probability of the test supply a negative result, since that the individual is free of the disease. In practice, it is usual to occur situations where a proportion of selected individuals cannot have verified the real state of the illness, to the fact that the verification test can be an invasive procedure, as in the diagnosis of lung cancer, or any other cases where risks are involved, therefore not practicable, nor ethical, or still procedures with high cost. Thus, instead of solve the problem, many studies of evaluation of performance of diagnostic tests are elaborated only using the information of verified individuals. This procedure can leads to biased results. This is known as verification bias, that consists of the calculation of estimates of sensitivity and specificity of diagnostic tests when only the individuals verified by the gold standard test are inserted in the analysis and the unverified ones, discarded or considered that they are free of the disease. This work presents a revision of the methodologies already proposed to calculate sensitivity and the specificity in the presence of the verifi cation bias, as well as a detailed analysis of the influence of the propor tion of individuals not verified, the effect of the sample size and the influ ence of choosing different prior densi ties, when using the bayesian methodo logy, in the calculation of these estima tes. It was also introduced a bayesian methodology to estimate performance measures of two diagnostic tests when the verification bias is present.

Neste trabalho foram utilizados modelos de regressão espaço-temporais bayesianos para estimar a incidência de TB em Ribeirão Preto (anos de 2006 a 2009) por área de abrangência de unidades de saúde, associando-a a covariáveis de interesse (IPVS, Renda e Educação predominantes naquelas áreas). O método baseia-se em simulações MCMC para estimar as distribuições a posteriori das incidências de TB em Ribeirão Preto. Como resultado, temos mapas que mostram mais claramente um padrão espacial, com estimativas mais suavizadas e com menos flutuações aleatórias. Observamos que as áreas com as mais altas taxas de incidência também possuem índice de vulnerabilidade social médio e alto. Em relação à renda, a faixa salarial predominante dos responsáveis pelo domicílio nestas regiões é entre 0 e 3 salários mínimos e o nível de escolaridade predominante dos chefes do domicílio nestas regiões é o ensino fundamental. Os resultados dos modelos bayesianos analisados nos evidenciam que com o aumento da vulnerabilidade social aumentamos significativamente a incidência de TB em Ribeirão Preto. Nas áreas onde a vulnerabilidade é alta a incidência de TB chega a quase 15 vezes a incidência das áreas sem vulnerabilidade. Houve um aumento significativo em relação à incidência de tuberculose em Ribeirão Preto durante os anos estudados, sendo as maiores incidências registradas no ano de 2009. O uso de mapas facilitou a visualização de áreas que merecem uma atenção especial no controle da TB, além disso, a associação da doença com renda, escolaridade e vulnerabilidade social trazem subsídios para que os gestores responsáveis pelo planejamento do município planejem intervenções com uma atenção especial a estas áreas, reunindo esforços para a redução da pobreza e da desigualdade social, alternativas para uma melhor distribuição de renda e melhorar o acesso ao saneamento básico dentre outras prioridades. / In this study we used Bayesian space-temporal regression models to estimate the incidence of TB in Ribeirão Preto, SP (years 2006 to 2009) by the coverage area of health units, associating it with the covariates of interest (IPVS, Income and Education predominant those areas). The method is based on MCMC simulations for estimate the posterior distributions of TB incidence in Ribeirão Preto. As a result, we have maps that show a spatial pattern more clearly, with estimates smoother and less random fluctuations. We observed that the areas with the highest incidence rates also have medium and high social vulnerability index. Concerning income, the prevailing salary range of household heads in these regions is between 0 and 3 minimum wages and the prevailing level of education of household heads in these regions is the elementary school. The results of the models in Bayesian analysis show that with increasing social vulnerability significantly increased the incidence of TB in Ribeirao Preto. In areas where vulnerability is high incidence of TB is nearly 15 times the incidence of areas without vulnerability. There was a significant increase in the incidence of tuberculosis in Ribeirão Preto during the years studied, the highest incidence recorded in 2009. The use of maps improved visualization of areas that deserve special attention for TB control, in addition, the association of disease with income, education and social vulnerability that bring benefits to the managers responsible for planning the municipality to plan interventions with special attention these areas, uniting efforts to reduce poverty and social inequality, alternatives to improve income distribution and improve access to basic sanitation among other priorities.

Nous abordons l'étude de deux problèmes inverses, le problème des moments de Hausdorff et celui de la déconvolution sur la sphère ainsi qu'un problème de régression en design aléatoire. Le problème des moments de Hausdorff consiste à estimer une densité de probabilité à partir d'une séquence de moments bruités. Nous établissons une borne supérieure pour notre estimateur ainsi qu'une borne inférieure pour la vitesse de convergence, démontrant ainsi que notre estimateur converge à la vitesse optimale pour les classes de régularité de type Sobolev. Quant au problème de déconvolution sur la sphère, nous proposons un nouvel algorithme qui combine la méthode SVD traditionnelle et une procédure de seuillage dans la base des Needlets sphériques. Nous donnons une borne supérieure en perte Lp et menons une étude numérique qui montre des résultats fort prometteurs. Le problème de la régression en design aléatoire est abordé sous le prisme bayésien et sur la base des ondelettes déformées. Nous considérons deux scenarios de modèles a priori faisant intervenir des gaussiennes à faible et à grande variance et fournissons des bornes supérieures pour l'estimateur de la médiane a posteriori. Nous menons aussi une étude numérique qui révèle de bonnes performances numériques.

<p>Urbanization causes myriad changes in watershed processes, ultimately disrupting the structure and function of stream ecosystems. Urban development introduces contaminants (human waste, pesticides, industrial chemicals). Impervious surfaces and artificial drainage systems speed the delivery of contaminants to streams, while bypassing soil filtration and local riparian processes that can mitigate the impacts of these contaminants, and disrupting the timing and volume of hydrologic patterns. Aquatic habitats where biota live are degraded by sedimentation, channel incision, floodplain disconnection, substrate alteration and elimination of reach diversity. These compounding changes ultimately lead to alteration of invertebrate community structure and function. Because the effects of urbanization on stream ecosystems are complex, multilayered, and interacting, modeling these effects presents many unique challenges, including: addressing and quantifying processes at multiple scales, representing major interrelated simultaneously acting dynamics at the system level, incorporating uncertainty resulting from imperfect knowledge, imperfect data, and environmental variability, and integrating multiple sources of available information about the system into the modeling construct. These challenges can be addressed by using a Bayesian modeling approach. Specifically, the use of multilevel hierarchical models and Bayesian network models allows the modeler to harness the hierarchical nature of the U.S. Geological Survey (USGS) Effect of Urbanization on Stream Ecosystems (EUSE) dataset to predict invertebrate response at both basin and regional levels, concisely represent and parameterize this system of complicated cause and effect relationships and uncertainties, calculate the full probabilistic function of all variables efficiently as the product of more manageable conditional probabilities, and includes both expert knowledge and data. Utilizing this Bayesian framework, this dissertation develops a series of statistically rigorous and ecologically interpretable models predicting the effect of urbanization on invertebrates, as well as a unique, systematic methodology that creates an informed expert prior and then updates this prior with available data using conjugate Dirichlet-multinomial distribution forms. The resulting models elucidate differences between regional responses to urbanization (particularly due to background agriculture and precipitation) and address the influences of multiple urban induced stressors acting simultaneously from a new system-level perspective. These Bayesian modeling approaches quantify previously unexplained regional differences in biotic response to urbanization, capture multiple interacting environmental and ecological processes affected by urbanization, and ultimately link urbanization effects on stream biota to a management context such that these models describe and quantify how changes in drivers lead to changes in regulatory endpoint (the Biological Condition Gradient; BCG).</p> / Dissertation

Global marketing managers are keenly interested in being able to predict the sales of their new products. Understanding how a product is adopted over time allows the managers to optimally allocate their resources. With the world becoming ever more global, there are strong and complex interactions between the countries in the world. My work explores how to describe the relationship between those countries and determines the best way to leverage that information to improve the sales predictions. In Chapter II, I describe how diffusion speed has changed over time. The most recent major study on this topic, by Christophe Van den Bulte, investigated new product di ffusions in the United States. Van den Bulte notes that a similar study is needed in the international context, especially in developing countries. Additionally, his model contains the implicit assumption that the diffusion speed parameter is constant throughout the life of a product. I model the time component as a nonparametric function, allowing the speed parameter the flexibility to change over time. I find that early in the product's life, the speed parameter is higher than expected. Additionally, as the Internet has grown in popularity, the speed parameter has increased. In Chapter III, I examine whether the interactions can be described through a reference hierarchy in addition to the cross-country word-of-mouth eff ects already in the literature. I also expand the word-of-mouth e ffect by relating the magnitude of the e ffect to the distance between the two countries. The current literature only applies that e ffect equally to the n closest countries (forming a neighbor set). This also leads to an analysis of how to best measure the distance between two countries. I compare four possible distance measures: distance between the population centroids, trade ow, tourism ow, and cultural similarity. Including the reference hierarchy improves the predictions by 30 percent over the current best model. Finally, in Chapter IV, I look more closely at the Bass Diffusion Model. It is prominently used in the marketing literature and is the base of my analysis in Chapter III. All of the current formulations include the implicit assumption that all the regression parameters are equal for each country. One dollar increase in GDP should have more of an eff ect in a poor country than in a rich country. A Dirichlet process prior enables me to cluster the countries by their regression coefficients. Incorporating the distance measures can improve the predictions by 35 percent in some cases.