[pt] Este trabalho modela a evolução temporal de uma epidemia com uma abordagem estocástica. O número de novas infecções por infectado é modelado como uma variável aleatória discreta, chamada aqui de contágio. Logo, a evolução temporal da doença é um processo estocástico. Mais especificamente, a propagação é dada pelo modelo de Bienaymé-Galton-Watson, um tipo de processo de ramificação de parâmetro discreto. Neste processo, para um determinado instante, o número de membros infectados, ou seja, a geração de membros infectados é uma variável aleatória. Na primeira parte da dissertação, dado que o modelo probabilístico do contágio é conhecido, quatro metodologias utilizadas para obter as funções de massa das gerações do processo estocástico são comparadas. As metodologias são: funções geradoras de probabilidade com e sem identidades polinomiais, cadeia de Markov e simulações de Monte Carlo. A primeira e terceira metodologias fornecem expressões analíticas relacionando a variável aleatória de contágio com a variável aleatória do tamanho de uma geração. Essas expressões analíticas são utilizadas na segunda parte desta dissertação, na qual o problema clássico de inferência paramétrica bayesiana é estudado. Com a ajuda do teorema de Bayes, parâmetros da variável aleatória de contágio são inferidos a partir de realizações do processo de ramificação. As expressões analíticas obtidas na primeira parte do trabalho são usadas para construir funções de verossimilhança apropriadas. Para resolver o problema inverso, duas maneiras diferentes de se usar dados provindos do processo de Bienaymé-Galton-Watson são desenvolvidas e comparadas: quando dados são realizações de uma única geração do processo de ramificação ou quando os dados são uma única realização do processo de ramificação observada ao longo de uma quantidade de gerações. O critério abordado neste trabalho para encerrar o processo de atualização na inferência paramétrica usa a distância de L2-Wasserstein, que é uma métrica baseada no transporte ótimo de massa. Todas as rotinas numéricas e simbólicas desenvolvidas neste trabalho são escritas em MATLAB. / [en] This work models an epidemic s spreading over time with a stochastic approach. The number of infections per infector is modeled as a discrete random variable, named here as contagion. Therefore, the evolution of the disease over time is a stochastic process. More specifically, this propagation is modeled as the Bienaymé-Galton-Watson process, one kind of branching process with discrete parameter. In this process, for a given time, the number of infected members, i.e. a generation of infected members, is a random variable. In the first part of this dissertation, given that the mass function of the contagion s random variable is known, four methodologies to find the mass function of the generations of the stochastic process are compared. The methodologies are: probability generating functions with and without polynomial identities, Markov chain and Monte Carlo simulations. The first and the third methodologies provide analytical expressions relating the contagion random variable and the generation s size random variable. These analytical expressions are used in the second part of this dissertation, where a classical inverse problem of bayesian parametric inference is studied. With the help of Bayes rule, parameters of the contagion random variable are inferred from realizations of the stochastic process. The analytical expressions obtained in the first part of the work are used to build appropriate likelihood functions. In order to solve the inverse problem, two different ways of using data from the Bienaymé-Galton-Watson process are developed and compared: when data are realizations of a single generation of the branching process and when data is just one realization of the branching process observed over a certain number of generations. The criteria used in this work to stop the update process in the bayesian parametric inference uses the L2-Wasserstein distance, which is a metric based on optimal mass transference. All numerical and symbolical routines developed to this work are written in MATLAB.

The time-course of interaction formation or breaking can be studied using LigandTracer, and the data obtained from an experiment can be analyzed using a model of ligand-target binding kinetics. There are different kinetic models, and the choice of model is currently motivated by knowledge about the interaction, which is problematic when the knowledge about the interaction is unsatisfactory. In this project, a Bayesian model selection procedure was implemented to motivate the model choice using the data obtained from studying a biological system. The model selection procedure was implemented for four kinetic models, the 1:1 model, the 1:2 model, the bivalent model and a new version of the bivalent model.Bayesian inference was performed on the data using each of the models to obtain the posterior distributions of the parameters. Afterwards, the Bayes factor was approximated from numerical calculations of the marginal likelihood. Four numerical methods were implemented to approximate the marginal likelihood, the Naïve Monte Carlo estimator, the method of Harmonic Means of the likelihood, Importance Sampling and Sequential Monte Carlo. When tested on simulated data, the method of Importance Sampling seemed to yield the most reliable prediction of the most likely model. The model selection procedure was then tested on experimental data which was expected to be from a 1:1 interaction and the result of the model selection procedure did not agree with the expectation on the experimental test dataset. Therefore no reliable conclusion could be made when the model selection procedure was used to analyze the interaction between the anti-CD20 antibody Rituximab and Daudi cells. / Interaktioner kan analyseras med hjälp av LigandTracer. Data från ett LigandTracer experiment kan sedan analyseras med avseende på en kinetisk modell. Det finns olika kinetiska modeller, och modellvalet motiveras vanligen utifrån tidigare kunskap om interaktionen, vilket är problematiskt när den tillgängliga informationen om en interaktion är otillräcklig. I det här projektet implementerades en Bayesiansk metod för att motivera valet av modell utifrån data från ett LigandTracer experiment. Modellvalsmetoden implementerades för fyra kinetiska modeller, 1:1 modellen, 1:2 modellen, den bivalenta modellen och en ny version av den bivalenta modellen. Bayesiansk inferens användes för att få fram aposteriorifördelningarna för de olika modellernas parametrar utifrån den givna datan. Sedan beräknades Bayes faktor utifrån numeriska approximationer av marginalsannolikeheten. Fyra numeriska metoder implementerades för att approximera marginalsannolikheten; Naïve Monte Carlo estimator, det harmoniska medelvärdet av likelihood-funktionen, Importance Sampling och Sekventiell Monte Carlo. När modellvalsmetoden testades på simulerad data gav metoden Importance Sampling den mest tillförlitliga förutsägelsen om vilken modell som generade datan. Metoden testades också på experimentell data som förväntades följa en 1:1 interaktion och resultatet avvek från det förväntade resultatet. Följaktligen kunde ingen slutsas dras av resultet från modelvalsmetoden när den sedan används för att analysera interaktionen mellan anti-CD antikroppen Rituximab och Daudi-celler.

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It remains a major challenge in biophysics to understand the conformational dynamics of biomolecules. As powerful tools, molecular dynamics (MD) simulations have become increasingly important in studying the full atomic details of conformational dynamics of biomolecules. In addition, many statistical models have been developed to give insight into the big datasets from MD simulations. In this work, I first describe three statistical models used to analyze MD simulation data: Lifson-Roig Helix-Coil theory, Bayesian inference models, and Markov state models. Then I present the applications of each model in analyzing MD simulations and revealing insight into the conformational dynamics of biomolecules. These statistical models allow us to bridge microscopic and macroscopic mechanisms of biological processes and connect simulations with experiments. / Chemistry

<p dir="ltr">It has been shown that formative assessments can lead to improvements in the learning process. Cognitive Diagnostic Models (CDMs) are a powerful formative assessment tool that can be used to provide individuals with valuable information regarding skill mastery in educational settings. These models provide each student with a ``skill mastery profile'' that shows the level of mastery they have obtained with regard to a specific set of skills. These profiles can be used to help both students and educators make more informed decisions regarding the educational process, which can in turn accelerate learning for students. However, despite their utility, these models are rarely used with small sample sizes. One reason for this is that these models are often complex, containing many parameters that can be difficult to estimate accurately when working with a small number of observations. This work aims to contribute to and expand upon previous work to make CDMs more accessible for a wider range of educators and students.</p><p dir="ltr">There are three main small sample statistical problems that we address in this work: 1) accurate estimation of the population distribution of skill mastery profiles, 2) accurate estimation of additional model parameters for CDMs as well as improved classification of individual skill mastery profiles, and 3) improved selection of an appropriate CDM for each item on the assessment. Each of these problems deals with a different aspect of educational measurement and the solutions provided to these problems can ultimately lead to improvements in the educational process for both students and teachers. By finding solutions to these problems that work well when using small sample sizes, we make it possible to improve learning in everyday classroom settings and not just in large scale assessment settings.</p><p dir="ltr">In the first part of this work, we propose novel algorithms for estimating the population distribution of skill mastery profiles for a popular CDM, the Deterministic Inputs Noisy ``and'' Gate (DINA) model. These algorithms borrow inspiration from the concepts behind popular machine learning algorithms. However, in contrast to these methods, which are often used solely for prediction, we illustrate how the ideas behind these methods can be adapted to obtain estimates of specific model parameters. Through studies involving simulated and real-life data, we illustrate how the proposed algorithms can be used to gain a better picture of the distribution of skill mastery profiles for an entire population students, but can do so by only using a small sample of students from that population. </p><p dir="ltr">In the second part of this work, we introduce a new method for regularizing high-dimensional CDMs using a class of Bayesian shrinkage priors known as catalytic priors. We show how a simpler model can first be fit to the observed data and then be used to generate additional pseudo-observations that, when combined with the original observations, make it easier to more accurately estimate the parameters in a complex model of interest. We propose an alternative, simpler model that can be used instead of the DINA model and show how the information from this model can be used to formulate an intuitive shrinkage prior that effectively regularizes model parameters. This makes it possible to improve the accuracy of parameter estimates for the more complex model, which in turn leads to better classification of skill mastery. We demonstrate the utility of this method in studies involving simulated and real-life data and show how the proposed approach is superior to other common approaches for small sample estimation of CDMs.</p><p dir="ltr">Finally, we discuss the important problem of selecting the most appropriate model for each item on assessment. Often, it is not uncommon in practice to use the same CDM for each item on an assessment. However, this can lead to suboptimal results in terms of parameter estimation and overall model fit. Current methods for item-level model selection rely on large sample asymptotic theory and are thus inappropriate when the sample size is small. We propose a Bayesian approach for performing item-level model selection using Reversible Jump Markov chain Monte Carlo. This approach allows for the simultaneous estimation of posterior probabilities and model parameters for each candidate model and does not require a large sample size to be valid. We again demonstrate through studies involving simulated and real-life data that the proposed approach leads to a much higher chance of selecting the best model for each item. This in turn leads to better estimates of item and other model parameters, which ultimately leads to more accurate information regarding skill mastery. </p>

Accurate modeling of structural ensembles is essential for understanding molecular function, predicting molecular interactions, refining molecular potentials, protein engineering, drug discovery, and more. Here, we enhance molecular modeling through Bayesian Inference of Conformational Populations (BICePs), a highly versatile algorithm for reweighting simulated ensembles with experimental data. By incorporating replica-averaging, improved likelihood functions to better address systematic errors, and adopting variational optimization schemes, the utility of this algorithm in the refinement and validation of both structural ensembles and empirical models is unmatched. Utilizing a set of diverse experimental measurements, including NOE distances, chemical shifts, and vicinal J-coupling constants, we evaluated nine force fields for simulating the mini-protein chignolin, highlighting BICePs’ capability to correctly identify folded conformations and perform objective model selection. Additionally, we demonstrate how BICePs automates the parameterization of molecular potentials and forward models—computational frameworks that generate observable quantities—while properly accounting for all sources of random and systematic error. By reconciling prior knowledge of structural ensembles with solution-based experimental observations, BICePs not only offers a robust approach for evaluating the predictive accuracy of molecular models but also shows significant promise for future applications in computational chemistry and biophysics. / Chemistry

L’imagerie hyperspectrale est très largement employée en télédétection pour diverses applications, dans le domaine civil comme dans le domaine militaire. Une image hyperspectrale est le résultat de l’acquisition d’une seule scène observée dans plusieurs longueurs d’ondes. Par conséquent, chacun des pixels constituant cette image est représenté par un vecteur de mesures (généralement des réflectances) appelé spectre. Une étape majeure dans l’analyse des données hyperspectrales consiste à identifier les composants macroscopiques (signatures) présents dans la région observée et leurs proportions correspondantes (abondances). Les dernières techniques développées pour ces analyses ne modélisent pas correctement ces images. En effet, habituellement ces techniques supposent l’existence de pixels purs dans l’image, c’est-à-dire des pixels constitué d’un seul matériau pur. Or, un pixel est rarement constitué d’éléments purs distincts l’un de l’autre. Ainsi, les estimations basées sur ces modèles peuvent tout à fait s’avérer bien loin de la réalité. Le but de cette étude est de proposer de nouveaux algorithmes d’estimation à l’aide d’un modèle plus adapté aux propriétés intrinsèques des images hyperspectrales. Les paramètres inconnus du modèle sont ainsi déduits dans un cadre Bayésien. L’utilisation de méthodes de Monte Carlo par Chaînes de Markov (MCMC) permet de surmonter les difficultés liées aux calculs complexes de ces méthodes d’estimation. / Hyperspectral imagery has been widely used in remote sensing for various civilian and military applications. A hyperspectral image is acquired when a same scene is observed at different wavelengths. Consequently, each pixel of such image is represented as a vector of measurements (reflectances) called spectrum. One major step in the analysis of hyperspectral data consists of identifying the macroscopic components (signatures) that are present in the sensored scene and the corresponding proportions (concentrations). The latest techniques developed for this analysis do not properly model these images. Indeed, these techniques usually assume the existence of pure pixels in the image, i.e. pixels containing a single pure material. However, a pixel is rarely composed of pure spectrally elements, distinct from each other. Thus, such models could lead to weak estimation performance. The aim of this thesis is to propose new estimation algorithms with the help of a model that is better suited to the intrinsic properties of hyperspectral images. The unknown model parameters are then infered within a Bayesian framework. The use of Markov Chain Monte Carlo (MCMC) methods allows one to overcome the difficulties related to the computational complexity of these inference methods.

Phylogenetic inference relies heavily on statistical models that have been extended and refined over the past years into complex hierarchical models to capture the intricacies of evolutionary processes. The wealth of information in the form of fully sequenced genomes has led to the development of methods that are used to reconstruct the gene and species evolutionary histories in greater and more accurate detail. However, genes are composed of evolutionary conserved sequence segments called domains, and domains can also be affected by duplications, losses, and bifurcations implied by gene or species evolution. This thesis proposes an extension of evolutionary models, such as duplication-loss, rate, and substitution, that have previously been used to model gene evolution, to model the domain evolution. In this thesis, I am proposing DomainDLRS: a comprehensive, hierarchical Bayesian method, based on the DLRS model by Åkerborg et al., 2009, that models domain evolution as occurring inside the gene and species tree. The method incorporates a birth-death process to model the domain duplications and losses along with a domain sequence evolution model with a relaxed molecular clock assumption. The method employs a variant of Markov Chain Monte Carlo technique called, Grouped Independence Metropolis-Hastings for the estimation of posterior distribution over domain and gene trees. By using this method, we performed analyses of Zinc-Finger and PRDM9 gene families, which provides an interesting insight of domain evolution. Finally, a synteny-aware approach for gene homology inference, called GenFamClust, is proposed that uses similarity and gene neighbourhood conservation to improve the homology inference. We evaluated the accuracy of our method on synthetic and two biological datasets consisting of Eukaryotes and Fungal species. Our results show that the use of synteny with similarity is providing a significant improvement in homology inference. / <p>QC 20160904</p>

The Bayesian approach to cluster analysis is presented. We assume that all data stem from a finite mixture model, where each component corresponds to one cluster and is given by a multivariate normal distribution with unknown mean and variance. The method produces posterior distributions of all cluster parameters and proportions as well as associated cluster probabilities for all objects. We extend this method in several directions to some common but non-standard situations. The first extension covers the case with a few deviant observations not belonging to one of the normal clusters. An extra component/cluster is created for them, which has a larger variance or a different distribution, e.g. is uniform over the whole range. The second extension is clustering of longitudinal data. All units are clustered at all time points separately and the movements between time points are modeled by Markov transition matrices. This means that the clustering at one time point will be affected by what happens at the neighbouring time points. The third extension handles datasets with missing data, e.g. item non-response. We impute the missing values iteratively in an extra step of the Gibbs sampler estimation algorithm. The Bayesian inference of mixture models has many advantages over the classical approach. However, it is not without computational difficulties. A software package, written in Matlab for Bayesian inference of mixture models is introduced. The programs of the package handle the basic cases of clustering data that are assumed to arise from mixture models of multivariate normal distributions, as well as the non-standard situations.