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<p>With the development of modern data collection approaches, researchers may collect hundreds to millions of variables, yet may not need to utilize all explanatory variables available in predictive models. Hence, choosing models that consist of a subset of variables often becomes a crucial step. In linear regression, variable selection not only reduces model complexity, but also prevents over-fitting. From a Bayesian perspective, prior specification of model parameters plays an important role in model selection as well as parameter estimation, and often prevents over-fitting through shrinkage and model averaging.</p><p>We develop two novel hierarchical priors for selection and model averaging, for Generalized Linear Models (GLMs) and normal linear regression, respectively. They can be considered as "spike-and-slab" prior distributions or more appropriately "spike- and-bell" distributions. Under these priors we achieve dimension reduction, since their point masses at zero allow predictors to be excluded with positive posterior probability. In addition, these hierarchical priors have heavy tails to provide robust- ness when MLE's are far from zero.</p><p>Zellner's g-prior is widely used in linear models. It preserves correlation structure among predictors in its prior covariance, and yields closed-form marginal likelihoods which leads to huge computational savings by avoiding sampling in the parameter space. Mixtures of g-priors avoid fixing g in advance, and can resolve consistency problems that arise with fixed g. For GLMs, we show that the mixture of g-priors using a Compound Confluent Hypergeometric distribution unifies existing choices in the literature and maintains their good properties such as tractable (approximate) marginal likelihoods and asymptotic consistency for model selection and parameter estimation under specific values of the hyper parameters.</p><p>While the g-prior is invariant under rotation within a model, a potential problem with the g-prior is that it inherits the instability of ordinary least squares (OLS) estimates when predictors are highly correlated. We build a hierarchical prior based on scale mixtures of independent normals, which incorporates invariance under rotations within models like ridge regression and the g-prior, but has heavy tails like the Zeller-Siow Cauchy prior. We find this method out-performs the gold standard mixture of g-priors and other methods in the case of highly correlated predictors in Gaussian linear models. We incorporate a non-parametric structure, the Dirichlet Process (DP) as a hyper prior, to allow more flexibility and adaptivity to the data.</p> / Dissertation

Dans cette thèse nous nous intéressons aux problèmes de la sélection de variables en régression linéaire. Ces travaux sont en particulier motivés par les développements récents en génomique, protéomique, imagerie biomédicale, traitement de signal, traitement d’image, en marketing, etc… Nous regardons ce problème selon les deux points de vue fréquentielle et bayésienne.Dans un cadre fréquentiel, nous proposons des méthodes pour faire face au problème de la sélection de variables, dans des situations pour lesquelles le nombre de variables peut être beaucoup plus grand que la taille de l’échantillon, avec présence possible d’une structure supplémentaire entre les variables, telle qu’une forte corrélation ou un certain ordre entre les variables successives. Les performances théoriques sont explorées ; nous montrons que sous certaines conditions de régularité, les méthodes proposées possèdent de bonnes propriétés statistiques, telles que des inégalités de parcimonie, la consistance au niveau de la sélection de variables et la normalité asymptotique.Dans un cadre bayésien, nous proposons une approche globale de la sélection de variables en régression construite sur les lois à priori g de Zellner dans une approche similaire mais non identique à celle de Liang et al. (2008) Notre choix ne nécessite aucune calibration. Nous comparons les approches de régularisation bayésienne et fréquentielle dans un contexte peu informatif où le nombre de variables est presque égal à la taille de l’échantillon. / We are interested in variable sélection in linear régression models. This research is motivated by recent development in microarrays, proteomics, brain images, among others. We study this problem in both frequentist and bayesian viewpoints.In a frequentist framework, we propose methods to deal with the problem of variable sélection, when the number of variables is much larger than the sample size with a possibly présence of additional structure in the predictor variables, such as high corrélations or order between successive variables. The performance of the proposed methods is theoretically investigated ; we prove that, under regularity conditions, the proposed estimators possess statistical good properties, such as Sparsity Oracle Inequalities, variable sélection consistency and asymptotic normality.In a Bayesian Framework, we propose a global noninformative approach for Bayesian variable sélection. In this thesis, we pay spécial attention to two calibration-free hierarchical Zellner’s g-priors. The first one is the Jeffreys prior which is not location invariant. A second one avoids this problem by only considering models with at least one variable in the model. The practical performance of the proposed methods is illustrated through numerical experiments on simulated and real world datasets, with a comparison betwenn Bayesian and frequentist approaches under a low informative constraint when the number of variables is almost equal to the number of observations.

Recent technological advances and remarkable successes have led to genome-wide association studies (GWAS) becoming a tool of choice for investigating the genetic basis of common complex human diseases. These studies typically involve samples from thousands of individuals, scanning their DNA at up to a million loci along the genome to discover genetic variants that affect disease risk. Hundreds of such variants are now known for common diseases, nearly all discovered by GWAS over the last three years. As a result, many new studies are planned for the future or are already underway. In this thesis, I present analysis results from actual studies and some developments in theory and methodology. The Wellcome Trust Case Control Consortium (WTCCC) published one of the first large-scale GWAS in 2007. I describe my contribution to this study and present the results from some of my follow-up analyses. I also present results from a GWAS of a bipolar disorder sub-phenotype, and a recent and on-going fine mapping experiment. Building on methods developed as part of the WTCCC, I describe a Bayesian approach to GWAS analysis and compare it to widely used frequentist approaches. I do so both theoretically, by interpreting each approach from the perspective of the other, and empirically, by comparing their performance in the context of replicated GWAS findings. I discuss the implications of these comparisons on the interpretation and analysis of GWAS generally, highlighting the advantages of the Bayesian approach. Finally, I examine the effect of linkage disequilibrium on the detection and estimation of various types of genetic effects, particularly non-additive effects. I derive a theoretical result showing how the power to detect a departure from an additive model at a marker locus decays faster than the power to detect an association.

We are interested in variable sélection in linear régression models. This research is motivated by recent development in microarrays, proteomics, brain images, among others. We study this problem in both frequentist and bayesian viewpoints.In a frequentist framework, we propose methods to deal with the problem of variable sélection, when the number of variables is much larger than the sample size with a possibly présence of additional structure in the predictor variables, such as high corrélations or order between successive variables. The performance of the proposed methods is theoretically investigated ; we prove that, under regularity conditions, the proposed estimators possess statistical good properties, such as Sparsity Oracle Inequalities, variable sélection consistency and asymptotic normality.In a Bayesian Framework, we propose a global noninformative approach for Bayesian variable sélection. In this thesis, we pay spécial attention to two calibration-free hierarchical Zellner's g-priors. The first one is the Jeffreys prior which is not location invariant. A second one avoids this problem by only considering models with at least one variable in the model. The practical performance of the proposed methods is illustrated through numerical experiments on simulated and real world datasets, with a comparison betwenn Bayesian and frequentist approaches under a low informative constraint when the number of variables is almost equal to the number of observations.

Cette thèse se décompose en deux parties. Dans un premier temps nous nous intéressons à la sélection bayésienne de variables dans un modèle probit mixte.L'objectif est de développer une méthode pour sélectionner quelques variables pertinentes parmi plusieurs dizaines de milliers tout en prenant en compte le design d'une étude, et en particulier le fait que plusieurs jeux de données soient fusionnés. Le modèle de régression probit mixte utilisé fait partie d'un modèle bayésien hiérarchique plus large et le jeu de données est considéré comme un effet aléatoire. Cette méthode est une extension de la méthode de Lee et al. (2003). La première étape consiste à spécifier le modèle ainsi que les distributions a priori, avec notamment l'utilisation de l'a priori conventionnel de Zellner (g-prior) pour le vecteur des coefficients associé aux effets fixes (Zellner, 1986). Dans une seconde étape, nous utilisons un algorithme Metropolis-within-Gibbs couplé à la grouping (ou blocking) technique de Liu (1994) afin de surmonter certaines difficultés d'échantillonnage. Ce choix a des avantages théoriques et computationnels. La méthode développée est appliquée à des jeux de données microarray sur le cancer du sein. Cependant elle a une limite : la matrice de covariance utilisée dans le g-prior doit nécessairement être inversible. Or il y a deux cas pour lesquels cette matrice est singulière : lorsque le nombre de variables sélectionnées dépasse le nombre d'observations, ou lorsque des variables sont combinaisons linéaires d'autres variables. Nous proposons donc une modification de l'a priori de Zellner en y introduisant un paramètre de type ridge, ainsi qu'une manière de choisir les hyper-paramètres associés. L'a priori obtenu est un compromis entre le g-prior classique et l'a priori supposant l'indépendance des coefficients de régression, et se rapproche d'un a priori précédemment proposé par Gupta et Ibrahim (2007).Dans une seconde partie nous développons deux nouvelles méthodes MCMC basées sur des populations de chaînes. Dans le cas de modèles complexes ayant de nombreux paramètres, mais où la vraisemblance des données peut se calculer, l'algorithme Equi-Energy Sampler (EES) introduit par Kou et al. (2006) est apparemment plus efficace que l'algorithme classique du Parallel Tempering (PT) introduit par Geyer (1991). Cependant, il est difficile d'utilisation lorsqu'il est couplé avec un échantillonneur de Gibbs, et nécessite un stockage important de valeurs. Nous proposons un algorithme combinant le PT avec le principe d'échanges entre chaînes ayant des niveaux d'énergie similaires dans le même esprit que l'EES. Cette adaptation appelée Parallel Tempering with Equi-Energy Moves (PTEEM) conserve l'idée originale qui fait la force de l'algorithme EES tout en assurant de bonnes propriétés théoriques et une utilisation facile avec un échantillonneur de Gibbs.Enfin, dans certains cas complexes l'inférence peut être difficile car le calcul de la vraisemblance des données s'avère trop coûteux, voire impossible. De nombreuses méthodes sans vraisemblance ont été développées. Par analogie avec le Parallel Tempering, nous proposons une méthode appelée ABC-Parallel Tempering, basée sur la théorie des MCMC, utilisant une population de chaînes et permettant des échanges entre elles. / This thesis is divided into two main parts. In the first part, we propose a Bayesian variable selection method for probit mixed models. The objective is to select few relevant variables among tens of thousands while taking into account the design of a study, and in particular the fact that several datasets are merged together. The probit mixed model used is considered as part of a larger hierarchical Bayesian model, and the dataset is introduced as a random effect. The proposed method extends a work of Lee et al. (2003). The first step is to specify the model and prior distributions. In particular, we use the g-prior of Zellner (1986) for the fixed regression coefficients. In a second step, we use a Metropolis-within-Gibbs algorithm combined with the grouping (or blocking) technique of Liu (1994). This choice has both theoritical and practical advantages. The method developed is applied to merged microarray datasets of patients with breast cancer. However, this method has a limit: the covariance matrix involved in the g-prior should not be singular. But there are two standard cases in which it is singular: if the number of observations is lower than the number of variables, or if some variables are linear combinations of others. In such situations we propose to modify the g-prior by introducing a ridge parameter, and a simple way to choose the associated hyper-parameters. The prior obtained is a compromise between the conditional independent case of the coefficient regressors and the automatic scaling advantage offered by the g-prior, and can be linked to the work of Gupta and Ibrahim (2007).In the second part, we develop two new population-based MCMC methods. In cases of complex models with several parameters, but whose likelihood can be computed, the Equi-Energy Sampler (EES) of Kou et al. (2006) seems to be more efficient than the Parallel Tempering (PT) algorithm introduced by Geyer (1991). However it is difficult to use in combination with a Gibbs sampler, and it necessitates increased storage. We propose an algorithm combining the PT with the principle of exchange moves between chains with same levels of energy, in the spirit of the EES. This adaptation which we are calling Parallel Tempering with Equi-Energy Move (PTEEM) keeps the original idea of the EES method while ensuring good theoretical properties and a practical use in combination with a Gibbs sampler.Then, in some complex models whose likelihood is analytically or computationally intractable, the inference can be difficult. Several likelihood-free methods (or Approximate Bayesian Computational Methods) have been developed. We propose a new algorithm, the Likelihood Free-Parallel Tempering, based on the MCMC theory and on a population of chains, by using an analogy with the Parallel Tempering algorithm.

CHAPTER 1:<p>The default g-priors predominant in Bayesian Model Averaging tend to over-concentrate posterior mass on a tiny set of models - a feature we denote as 'supermodel effect'. To address it, we propose a 'hyper-g' prior specification, whose data-dependent shrinkage adapts posterior model distributions to data quality. We demonstrate the asymptotic consistency of the hyper-g prior, and its interpretation as a goodness-of-fit indicator. Moreover, we highlight the similarities between hyper-g and 'Empirical Bayes' priors, and introduce closed-form expressions essential to computationally feasibility. The robustness of the hyper-g prior is demonstrated via simulation analysis, and by comparing four vintages of economic growth data.<p><p>CHAPTER 2:<p>Ciccone and Jarocinski (2010) show that inference in Bayesian Model Averaging (BMA) can be highly sensitive to small data perturbations. In particular they demonstrate that the importance attributed to potential growth determinants varies tremendously over different revisions of international income data. They conclude that 'agnostic' priors appear too sensitive for this strand of growth empirics. In response, we show that the found instability owes much to a specific BMA set-up: First, comparing the same countries over data revisions improves robustness. Second, much of the remaining variation can be reduced by applying an evenly 'agnostic', but flexible prior.<p><p>CHAPTER 3:<p>This chapter explores the link between the leverage of the US financial sector, of households and of non-financial businesses, and real activity. We document that leverage is negatively correlated with the future growth of real activity, and positively linked to the conditional volatility of future real activity and of equity returns. <p>The joint information in sectoral leverage series is more relevant for predicting future real activity than the information contained in any individual leverage series. Using in-sample regressions and out-of sample forecasts, we show that the predictive power of leverage is roughly comparable to that of macro and financial predictors commonly used by forecasters. <p>Leverage information would not have allowed to predict the 'Great Recession' of 2008-2009 any better than conventional macro/financial predictors. <p><p>CHAPTER 4:<p>Model averaging has proven popular for inference with many potential predictors in small samples. However, it is frequently criticized for a lack of robustness with respect to prediction and inference. This chapter explores the reasons for such robustness problems and proposes to address them by transforming the subset of potential 'control' predictors into principal components in suitable datasets. A simulation analysis shows that this approach yields robustness advantages vs. both standard model averaging and principal component-augmented regression. Moreover, we devise a prior framework that extends model averaging to uncertainty over the set of principal components and show that it offers considerable improvements with respect to the robustness of estimates and inference about the importance of covariates. Finally, we empirically benchmark our approach with popular model averaging and PC-based techniques in evaluating financial indicators as alternatives to established macroeconomic predictors of real economic activity. / Doctorat en Sciences économiques et de gestion / info:eu-repo/semantics/nonPublished