We develop statistical methods for tackling two important problems in genetic association studies. First, we propose a Bayesian approach to overcome the winner's curse in genetic studies. Second, we consider a Bayesian latent variable model for analyzing longitudinal family data with pleiotropic phenotypes. Winner's curse in genetic association studies refers to the estimation bias of the reported odds ratios (OR) for an associated genetic variant from the initial discovery samples. It is a consequence of the sequential procedure in which the estimated effect of an associated genetic marker must first pass a stringent significance threshold. We propose a hierarchical Bayes method in which a spike-and-slab prior is used to account for the possibility that the significant test result may be due to chance. We examine the robustness of the method using different priors corresponding to different degrees of confidence in the testing results and propose a Bayesian model averaging procedure to combine estimates produced by different models. The Bayesian estimators yield smaller variance compared to the conditional likelihood estimator and outperform the latter in the low power studies. We investigate the performance of the method with simulations and applications to four real data examples. Pleiotropy occurs when a single genetic factor influences multiple quantitative or qualitative phenotypes, and it is present in many genetic studies of complex human traits. The longitudinal family studies combine the features of longitudinal studies in individuals and cross-sectional studies in families. Therefore, they provide more information about the genetic and environmental factors associated with the trait of interest. We propose a Bayesian latent variable modeling approach to model multiple phenotypes simultaneously in order to detect the pleiotropic effect and allow for longitudinal and/or family data. An efficient MCMC algorithm is developed to obtain the posterior samples by using hierarchical centering and parameter expansion techniques. We apply spike and slab prior methods to test whether the phenotypes are significantly associated with the latent disease status. We compute Bayes factors using path sampling and discuss their application in testing the significance of factor loadings and the indirect fixed effects. We examine the performance of our methods via extensive simulations and apply them to the blood pressure data from a genetic study of type 1 diabetes (T1D) complications.

We develop statistical methods for tackling two important problems in genetic association studies. First, we propose a Bayesian approach to overcome the winner's curse in genetic studies. Second, we consider a Bayesian latent variable model for analyzing longitudinal family data with pleiotropic phenotypes. Winner's curse in genetic association studies refers to the estimation bias of the reported odds ratios (OR) for an associated genetic variant from the initial discovery samples. It is a consequence of the sequential procedure in which the estimated effect of an associated genetic marker must first pass a stringent significance threshold. We propose a hierarchical Bayes method in which a spike-and-slab prior is used to account for the possibility that the significant test result may be due to chance. We examine the robustness of the method using different priors corresponding to different degrees of confidence in the testing results and propose a Bayesian model averaging procedure to combine estimates produced by different models. The Bayesian estimators yield smaller variance compared to the conditional likelihood estimator and outperform the latter in the low power studies. We investigate the performance of the method with simulations and applications to four real data examples. Pleiotropy occurs when a single genetic factor influences multiple quantitative or qualitative phenotypes, and it is present in many genetic studies of complex human traits. The longitudinal family studies combine the features of longitudinal studies in individuals and cross-sectional studies in families. Therefore, they provide more information about the genetic and environmental factors associated with the trait of interest. We propose a Bayesian latent variable modeling approach to model multiple phenotypes simultaneously in order to detect the pleiotropic effect and allow for longitudinal and/or family data. An efficient MCMC algorithm is developed to obtain the posterior samples by using hierarchical centering and parameter expansion techniques. We apply spike and slab prior methods to test whether the phenotypes are significantly associated with the latent disease status. We compute Bayes factors using path sampling and discuss their application in testing the significance of factor loadings and the indirect fixed effects. We examine the performance of our methods via extensive simulations and apply them to the blood pressure data from a genetic study of type 1 diabetes (T1D) complications.

This research aims to integrate mathematical water quality models with Bayesian inference techniques for obtaining effective model calibration and rigorous assessment of the uncertainty underlying model predictions. The first part of my work combines a Bayesian calibration framework with a complex biogeochemical model to reproduce oligo-, meso- and eutrophic lake conditions. The model accurately describes the observed patterns and also provides realistic estimates of predictive uncertainty for water quality variables. The Bayesian estimations are also used for appraising the exceedance frequency and confidence of compliance of different water quality criteria. The second part introduces a Bayesian hierarchical framework (BHF) for calibrating eutrophication models at multiple systems (or sites of the same system). The models calibrated under the BHF provided accurate system representations for all the scenarios examined. The BHF allows overcoming problems of insufficient local data by “borrowing strength” from well-studied sites. Both frameworks can facilitate environmental management decisions.

We considered a Bayesian approach to nonlinear inverse problems in which the unknown quantity is a high dimension spatial field. The Bayesian approach contains a natural mechanism for regularization in the form of prior information, can incorporate information from heterogeneous sources and provides a quantitative assessment of uncertainty in the inverse solution. The Bayesian setting casts the inverse solution as a posterior probability distribution over the model parameters. Karhunen-Lo'eve expansion and Discrete Cosine transform were used for dimension reduction of the random spatial field. Furthermore, we used a hierarchical Bayes model to inject multiscale data in the modeling framework. In this Bayesian framework, we have shown that this inverse problem is well-posed by proving that the posterior measure is Lipschitz continuous with respect to the data in total variation norm. The need for multiple evaluations of the forward model on a high dimension spatial field (e.g. in the context of MCMC) together with the high dimensionality of the posterior, results in many computation challenges. We developed two-stage reversible jump MCMC method which has the ability to screen the bad proposals in the first inexpensive stage. Channelized spatial fields were represented by facies boundaries and variogram-based spatial fields within each facies. Using level-set based approach, the shape of the channel boundaries was updated with dynamic data using a Bayesian hierarchical model where the number of points representing the channel boundaries is assumed to be unknown. Statistical emulators on a large scale spatial field were introduced to avoid the expensive likelihood calculation, which contains the forward simulator, at each iteration of the MCMC step. To build the emulator, the original spatial field was represented by a low dimensional parameterization using Discrete Cosine Transform (DCT), then the Bayesian approach to multivariate adaptive regression spline (BMARS) was used to emulate the simulator. Various numerical results were presented by analyzing simulated as well as real data.

This thesis, which consists of four chapters, focuses on forecasting in a data-rich environment and related computational issues. Chapter 1, “An embarrassment of riches: Forecasting using large panels” explores the idea of combining forecasts from various indicator models by using Bayesian model averaging (BMA) and compares the predictive performance of BMA with predictive performance of factor models. The combination of these two methods is also implemented, together with a benchmark, a simple autoregressive model. The forecast comparison is conducted in a pseudo out-of-sample framework for three distinct datasets measured at different frequencies. These include monthly and quarterly US datasets consisting of more than 140 predictors, and a quarterly Swedish dataset with 77 possible predictors. The results show that none of the considered methods is uniformly superior and that no method consistently outperforms or underperforms a simple autoregressive process. Chapter 2. “Forecast combination using predictive measures” proposes using out-of-sample predictive likelihood as the basis for BMA and forecast combination. In addition to its intuitive appeal, the use of the predictive likelihood relaxes the need to specify proper priors for the parameters of each model. We show that the forecast weights based on the predictive likelihood have desirable asymptotic properties. And that these weights will have better small sample properties than the traditional in-sample marginal likelihood when uninformative priors are used. In order to calculate the weights for the combined forecast, a number of observations, a hold-out sample, is needed. There is a trade off involved in the size of the hold-out sample. The number of observations available for estimation is reduced, which might have a detrimental effect. On the other hand, as the hold-out sample size increases, the predictive measure becomes more stable and this should improve performance. When there is a true model in the model set, the predictive likelihood will select the true model asymptotically, but the convergence to the true model is slower than for the marginal likelihood. It is this slower convergence, coupled with protection against overfitting, which is the reason the predictive likelihood performs better when the true model is not in the model set. In Chapter 3. “Forecasting GDP with factor models and Bayesian forecast combination” the predictive likelihood approach developed in the previous chapter is applied to forecasting GDP growth. The analysis is performed on quarterly economic dataset from six countries: Canada, Germany, Great Britain, Italy, Japan and United States. The forecast combination technique based on both in-sample and out-of-sample weights is compared to forecasts based on factor models. The traditional point forecast analysis is extended by considering confidence intervals. The results indicate that forecast combinations based on the predictive likelihood weights have better forecasting performance compared with the factor models and forecast combinations based on the traditional in-sample weights. In contrast to common findings, the predictive likelihood does improve upon an autoregressive process for longer horizons. The largest improvement over the in-sample weights is for small values of hold-out sample sizes, which provides protection against structural breaks at the end of the sample period. The potential benefits of model averaging as a tool for extracting the relevant information from a large set of predictor variables come at the cost of considerable computational complexity. To avoid evaluating all the models, several approaches have been developed to simulate from the posterior distributions. Markov chain Monte Carlo methods can be used to directly draw from the model posterior distributions. It is desirable that the chain moves well through the model space and takes draws from regions with high probabilities. Several computationally efficient sampling schemes, either one at a time or in blocks, have been proposed for speeding up convergence. There is a trade-off between local moves, which make use of the current parameter values to propose plausible values for model parameters, and more global transitions, which potentially allow faster exploration of the distribution of interest, but may be much harder to implement efficiently. Local model moves enable use of fast updating schemes, where it is unnecessary to completely reestimate the new, slightly modified, model to obtain an updated solution. The last fourth chapter “Computational efficiency in Bayesian model and variable selection” investigates the possibility of increasing computational efficiency by using alternative algorithms to obtain estimates of model parameters as well as keeping track of their numerical accuracy. Also, various samplers that explore the model space are presented and compared based on the output of the Markov chain. / Diss. Stockholm : Handelshögskolan, 2006

The main objective of this thesis is to develop novel Monte Carlo techniques with emphasis on various applications in finance and economics, particularly in the fields of risk management and asset returns modeling. New stochastic algorithms are developed for rare-event probability estimation, combinatorial optimization, parameter estimation and model selection. The contributions of this thesis are fourfold. Firstly, we study an NP-hard combinatorial optimization problem, the Winner Determination Problem (WDP) in combinatorial auctions, where buyers can bid on bundles of items rather than bidding on them sequentially. We present two randomized algorithms, namely, the cross-entropy (CE) method and the ADAptive Mulitilevel splitting (ADAM) algorithm, to solve two versions of the WDP. Although an efficient deterministic algorithm has been developed for one version of the WDP, it is not applicable for the other version considered. In addition, the proposed algorithms are straightforward and easy to program, and do not require specialized software. Secondly, two major applications of conditional Monte Carlo for estimating rare-event probabilities are presented: a complex bridge network reliability model and several generalizations of the widely popular normal copula model used in managing portfolio credit risk. We show how certain efficient conditional Monte Carlo estimators developed for simple settings can be extended to handle complex models involving hundreds or thousands of random variables. In particular, by utilizing an asymptotic description on how the rare event occurs, we derive algorithms that are not only easy to implement, but also compare favorably to existing estimators. Thirdly, we make a contribution at the methodological front by proposing an improvement of the standard CE method for estimation. The improved method is relevant, as recent research has shown that in some high-dimensional settings the likelihood ratio degeneracy problem becomes severe and the importance sampling estimator obtained from the CE algorithm becomes unreliable. In contrast, the performance of the improved variant does not deteriorate as the dimension of the problem increases. Its utility is demonstrated via a high-dimensional estimation problem in risk management, namely, a recently proposed t-copula model for credit risk. We show that even in this high-dimensional model that involves hundreds of random variables, the proposed method performs remarkably well, and compares favorably to existing importance sampling estimators. Furthermore, the improved CE algorithm is then applied to estimating the marginal likelihood, a quantity that is fundamental in Bayesian model comparison and Bayesian model averaging. We present two empirical examples to demonstrate the proposed approach. The first example involves women's labor market participation and we compare three different binary response models in order to find the one best fits the data. The second example utilizes two vector autoregressive (VAR) models to analyze the interdependence and structural stability of four U.S. macroeconomic time series: GDP growth, unemployment rate, interest rate, and inflation. Lastly, we contribute to the growing literature of asset returns modeling by proposing several novel models that explicitly take into account various recent findings in the empirical finance literature. Specifically, two classes of stylized facts are particularly important. The first set is concerned with the marginal distributions of asset returns. One prominent feature of asset returns is that the tails of their distributions are heavier than those of the normal---large returns (in absolute value) occur much more frequently than one might expect from a normally distributed random variable. Another robust empirical feature of asset returns is skewness, where the tails of the distributions are not symmetric---losses are observed more frequently than large gains. The second set of stylized facts is concerned with the dependence structure among asset returns. Recent empirical studies have cast doubts on the adequacy of the linear dependence structure implied by the multivariate normal specification. For example, data from various asset markets, including equities, currencies and commodities markets, indicate the presence of extreme co-movement in asset returns, and this observation is again incompatible with the usual assumption that asset returns are jointly normally distributed. In light of the aforementioned empirical findings, we consider various novel models that generalize the usual normal specification. We develop efficient Markov chain Monte Carlo (MCMC) algorithms to estimate the proposed models. Moreover, since the number of plausible models is large, we perform a formal Bayesian model comparison to determine the model that best fits the data. In this way, we can directly compare the two approaches of modeling asset returns: copula models and the joint modeling of returns.

[pt] Constituído por uma matriz eletro-energética predominantemente hídrica e território de proporções continentais, o Brasil apresenta características únicas, sendo possível realizar o aproveitamento dos fartos recursos hídricos presentes no território nacional. Aproximadamente 65 por cento da capacidade de geração de energia elétrica advém de recursos hidrelétricos enquanto 28 por cento de recursos termelétricos. Sabe-se que regimes hidrológicos de vazões naturais são de natureza estocástica e em função disso é preciso tratá-los para que se possa planejar a operação do sistema, sendo assim, o despacho hidrotérmico é de suma importância e caracterizado por sua dependência estocástica. A partir das vazões naturais é possível calcular a Energia Natural Afluente (ENA) que será utilizada diretamente no processo de simulação de séries sintéticas que, por sua vez, são utilizadas no processo de otimização, responsável pelo cálculo da política ótima visando minimizar os custos de operação do sistema. Os estudos referentes a simulação de cenários sintéticos de ENA vêm se desenvolvendo com novas propostas metodológicas ao longo dos anos. Tais desenvolvimentos muitas vezes pressupõem Gaussianidade dos dados, de forma que seja possível ajustar uma distribuição paramétrica nos mesmos. Percebeu-se que na maioria dos casos reais, no contexto do Setor Elétrico Brasileiro, os dados não podem ser tratados desta forma, uma vez que apresentam em sua densidade comportamentos de cauda relevantes e uma acentuada assimetria. É necessário para o planejamento da operação do Sistema Interligado Nacional (SIN) que a assimetria intrínseca a este comportamento seja passível de reprodução. Dessa forma, este trabalho propõe duas abordagens não paramétricas para simulação de cenários. A primeira refere-se ao processo de amostragem dos resíduos das séries de ENA, para tanto, utiliza-se a técnica Markov Chain Monte Carlo (MCMC) e o Kernel Density Estimation. A segunda metodologia proposta aplica o MCMC Interconfigurações diretamente nas séries de ENA para simulação de cenários sintéticos a partir de uma abordagem inovadora para transição entre as matrizes e períodos. Os resultados da implementação das metodologias, observados graficamente e a partir de testes estatísticos de aderência ao histórico de dados, apontam que as propostas conseguem reproduzir com uma maior acurácia as características assimétricas sem perder a capacidade de reproduzir estatísticas básicas. Destarte, pode-se afirmar que os modelos propostos são boas alternativas em relação ao modelo vigente utilizado pelo setor elétrico brasileiro. / [en] Consisting of an electro-energetic matrix with hydro predominance and a continental proportion territory, Brazil presents unique characteristics, being able to make use of the abundant water resources in the national territory. Approximately 65 percent of the electricity generation capacity comes from hydropower while 28 percent from thermoelectric plants. It is known that hydrological regimes have a stochastic nature and it is necessary to treat them so the energy system can be planned, thus the hydrothermal dispatch is extremely important and characterized by its stochastic dependence. From the natural streamflows it is possible to calculate the Natural Inflow Energy (NIE) that will be used directly in the synthetic series simulation process, which, in turn, are used on the optimization process, responsible for optimal policy calculation in order to minimize the system operational costs. The studies concerning the simulation of synthetic scenarios of NIE have been developing with new methodological proposals over the years. Such developments often presuppose data Gaussianity, so that a parametric distribution can be fitted to them. It was noticed that in the majority of real cases, in the context of the Brazilian Electrical Sector, the data cannot be treated like that, since they present in their density relevant tail behavior and skewness. It is necessary for the National Interconnected System (SIN) operational planning that the intrinsic skewness behavior is amenable to reproduction. Thus, this paper proposes two non-parametric approaches to scenarios simulation. The first one refers to the process of NIE series residues sampling, using a Markov Chain Monte Carlo (MCMC) technique and the Kernel Density Estimation. The second methodology is also proposed where the MCMC is applied periodically and directly in the NIE series to simulate synthetic scenarios using an innovative approach for transitions between matrices. The methodologies implementation results, observed graphically and based on statistical tests of adherence to the historical data, indicate that the proposals can reproduce with greater accuracy the asymmetric characteristics without losing the ability to reproduce basic statistics. Thus, one can conclude that the proposed models are good alternatives in relation to the current model of the Brazilian Electric Sector.

This thesis explores ideas from transport theory and optimal control to develop novel Monte Carlo methods to perform efficient statistical computation. The first project considers the problem of constructing a transport map between two given probability measures. In the Bayesian formalism, this approach is natural when one introduces a curve of probability measures connecting the prior to posterior by tempering the likelihood function. The main idea is to move samples from the prior using an ordinary differential equation (ODE), constructed by solving the Liouville partial differential equation (PDE) which governs the time evolution of measures along the curve. In this work, we first study the regularity solutions of Liouville equation should satisfy to guarantee validity of this construction. We place an emphasis on understanding these issues as it explains the difficulties associated with solutions that have been previously reported. After ensuring that the flow transport problem is well-defined, we give a constructive solution. However, this result is only formal as the representation is given in terms of integrals which are intractable. For computational tractability, we proposed a novel approximation of the PDE which yields an ODE whose drift depends on the full conditional distributions of the intermediate distributions. Even when the ODE is time-discretized and the full conditional distributions are approximated numerically, the resulting distribution of mapped samples can be evaluated and used as a proposal within Markov chain Monte Carlo and sequential Monte Carlo (SMC) schemes. We then illustrate experimentally that the resulting algorithm can outperform state-of-the-art SMC methods at a fixed computational complexity. The second project aims to exploit ideas from optimal control to design more efficient SMC methods. The key idea is to control the proposal distribution induced by a time-discretized Langevin dynamics so as to minimize the Kullback-Leibler divergence of the extended target distribution from the proposal. The optimal value functions of the resulting optimal control problem can then be approximated using algorithms developed in the approximate dynamic programming (ADP) literature. We introduce a novel iterative scheme to perform ADP, provide a theoretical analysis of the proposed algorithm and demonstrate that the latter can provide significant gains over state-of-the-art methods at a fixed computational complexity.

Nous avons développé une nouvelle approche basée sur les méthodes Monte Carlo par chaînes de Markov pour déterminer les distributions de Partons et quantifier leurs incertitudes expérimentales. L’intérêt principal d’une telle étude repose sur la possibilité de remplacer la minimisation standard avec MINUIT de la fonction χ 2 par des procédures fondées sur les méthodes Statistiques et sur l’inférence Bayésienne en particulier,offrant ainsi une meilleure compréhension de la détermination des distributions de partons. Après avoir examiné ces techniques Monte Carlo par chaînes de Markov, nous introduisons l’algorithme que nous avons choisi de mettre en œuvre, à savoir le Monte Carlo hybride (ou Hamiltonien). Cet algorithme, développé initialement pour la chromodynamique quantique sur réseau, s’avère très intéressant lorsqu’il est appliqué à la détermination des distributions de partons par des analyses globales. Nous avons montré qu’il permet de contourner les difficultés techniques dues à la grande dimensionnalité du problème, en particulier celle relative au taux d’acceptation. L’étude de faisabilité réalisée et présentée dans cette thèse indique que la méthode Monte Carlo par chaînes de Markov peut être appliquée avec succès à l’extraction des distributions de partons et à leurs in-certitudes expérimentales. / We have developed a new approach to determine parton distribution functions and quantify their experimental uncertainties, based on Markov Chain Monte Carlo methods.The main interest devoted to such a study is that we can replace the standard χ 2 MINUIT minimization by procedures grounded on Statistical Methods, and on Bayesian inference in particular, thus offering additional insight into the rich field of PDFs determination.After reviewing these Markov chain Monte Carlo techniques, we introduce the algorithm we have chosen to implement – namely Hybrid (or Hamiltonian) Monte Carlo. This algorithm, initially developed for lattice quantum chromodynamique, turns out to be very interesting when applied to parton distribution functions determination by global analyses ; we have shown that it allows to circumvent the technical difficulties due to the high dimensionality of the problem, in particular concerning the acceptance rate. The feasibility study performed and presented in this thesis, indicates that Markov chain Monte Carlo method can successfully be applied to the extraction of PDFs and of their experimental uncertainties.

La thèse traite du problème de l’échantillonnage gaussien en grande dimension.Un tel problème se pose par exemple dans les problèmes inverses bayésiens en imagerie où le nombre de variables atteint facilement un ordre de grandeur de 106_109.La complexité du problème d’échantillonnage est intrinsèquement liée à la structure de la matrice de covariance. Pour résoudre ce problème différentes solutions ont déjà été proposées,parmi lesquelles nous soulignons l’algorithme de Hogwild qui exécute des mises à jour de Gibbs locales en parallèle avec une synchronisation globale périodique.Notre algorithme utilise la connexion entre une classe d’échantillonneurs itératifs et les solveurs itératifs pour les systèmes linéaires. Il ne cible pas la distribution gaussienne requise, mais cible une distribution approximative. Cependant, nous sommes en mesure de contrôler la disparité entre la distribution approximative est la distribution requise au moyen d’un seul paramètre de réglage.Nous comparons d’abord notre algorithme avec les algorithmes de Gibbs et Hogwild sur des problèmes de taille modérée pour différentes distributions cibles. Notre algorithme parvient à surpasser les algorithmes de Gibbs et Hogwild dans la plupart des cas. Notons que les performances de notre algorithme dépendent d’un paramètre de réglage.Nous comparons ensuite notre algorithme avec l’algorithme de Hogwild sur une application réelle en grande dimension, à savoir la déconvolution-interpolation d’image.L’algorithme proposé permet d’obtenir de bons résultats, alors que l’algorithme de Hogwild ne converge pas. Notons que pour des petites valeurs du paramètre de réglage, notre algorithme ne converge pas non plus. Néanmoins, une valeur convenablement choisie pour ce paramètre permet à notre échantillonneur de converger et d’obtenir de bons résultats. / The thesis deals with the problem of high-dimensional Gaussian sampling.Such a problem arises for example in Bayesian inverse problems in imaging where the number of variables easily reaches an order of 106_109. The complexity of the sampling problem is inherently linked to the structure of the covariance matrix. Different solutions to tackle this problem have already been proposed among which we emphasizethe Hogwild algorithm which runs local Gibbs sampling updates in parallel with periodic global synchronisation.Our algorithm makes use of the connection between a class of iterative samplers and iterative solvers for systems of linear equations. It does not target the required Gaussian distribution, instead it targets an approximate distribution. However, we are able to control how far off the approximate distribution is with respect to the required one by means of asingle tuning parameter.We first compare the proposed sampling algorithm with the Gibbs and Hogwild algorithms on moderately sized problems for different target distributions. Our algorithm manages to out perform the Gibbs and Hogwild algorithms in most of the cases. Let us note that the performances of our algorithm are dependent on the tuning parameter.We then compare the proposed algorithm with the Hogwild algorithm on a large scalereal application, namely image deconvolution-interpolation. The proposed algorithm enables us to obtain good results, whereas the Hogwild algorithm fails to converge. Let us note that for small values of the tuning parameter our algorithm fails to converge as well.Not with standing, a suitably chosen value for the tuning parameter enables our proposed sampler to converge and to deliver good results.