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Probabilistic models are dominant in many research areas. To learn those models we need to find a way to determine parameters of distributions over variables which are included in the model. The main focus of my research is related to continuous variables. Thus, Gaussian distribution over variables is the most dominant factor in all models used in this document. I have been working on different and important real-life problems such as Uncertainty of Neural Network Based Aerosol Retrieval, Regression Learning with Multiple Noise Oracles and Model Predictive Control (MPC) for Sepsis Treatment, Clustering Causes of Action in Federal Courts. These problems will be discussed in the following chapters. Aerosols, small particles emanating from natural and man-made sources, along with green house gases have been recognized as very important factors in ongoing climate changes. Accurate estimation of aerosol composition and concentration is one of the main challenges in current climate research. Algorithm for prediction of aerosol designed by domain scientists does not provide quantitative information about aerosol estimation uncertainty. We deployed algorithm which uses neural networks to determine both uncertainty and the estimation of the aerosol. The uncertainty estimator has been built under an assumption that uncertainty is a function of variables used for aerosol prediction. Also, the uncertainty of predictions has been computed as the variance of the conditional distribution of targets given the input data. In regression learning, it is often difficult to obtain the true values of the label variables, while multiple sources of noisy estimates of lower quality are readily available. To address this problem, I propose a new Bayesian approach that learns a regression model from a data with noisy labels which are provided by multiple oracles. This method gives closed form solution for model parameters and it is applicable to both linear and nonlinear regression problems. Sepsis is a medical condition characterized as a systemic inflammatory response to an infection. High mortality rate (30-35%) of septic patients is usually caused by inadequate treatment. Thus, development of tools that can aid clinicians in designing optimal strategies for inflammation treatments is of utmost importance. Towards this objective I developed a data driven approach for therapy optimization where a predictive model for patients' behavior is learned directly from historical data. As such, the predictive model is incorporated into a model predictive control optimization algorithm to find optimal therapy, which will lead the patient to a healthy state. A more careful targeting of specific therapeutic strategies to more biologically homogeneous groups of patients is essential to developing effective sepsis treatment. We propose a kernel-based approach to characterize dynamics of inflammatory response in a heterogeneous population of septic patients. The method utilizes Linear State Space Control (LSSC) models to take into account dynamics of inflammatory response over time as well as the effect of therapy applied to the patient. We use a similarity measure defined on kernels of LSSC models to find homogeneous groups of patients. In addition to clustering of dynamics of inflammatory response we also explored a clustering of civil litigation from its inception by examining the content of civil complaints. We utilize spectral cluster analysis on a newly compiled federal district court dataset of causes of action in complaints to illustrate the relationship of legal claims to one another, the broader composition of lawsuits in trial courts, and the breadth of pleading in individual complaints. Our results shed light not only on the networks of legal theories in civil litigation but also on how lawsuits are classified and the strategies that plaintiffs and their attorneys employ when commencing litigation. / Computer and Information Science

Yes / Lorenz values and the Gini index are popular quantities in Mathematical Economics, and are used here in the context of quantum systems with finite-dimensional Hilbert space. They quantify the uncertainty in the probability distribution related to an orthonormal basis. It is shown that Lorenz values are superadditive functions and the Gini indices are subadditive functions. The supremum over all density matrices of the sum of the two Gini indices with respect to position and momentum states is used to define an uncertainty coefficient which quantifies the uncertainty in the quantum system. It is shown that the uncertainty coefficient is positive, and an upper bound for it is given. Various examples demonstrate these ideas.

This dissertation develops a new method of modeling uncertainty with spatial data called Bayesian spatial predictive synthesis (BSPS) and compares its predictive accuracy to established methods. Spatial data are often non-linear, complex, and difficult to capture with a single model. Existing methods such as model selection or simple model ensembling fail to consider the critical spatially varying model uncertainty problem; different models perform better or worse in different regions. BSPS can capture the model uncertainty by specifying a latent factor coefficient model that varies spatially as a synthesis function. This allows the model coefficients to vary across a region to achieve flexible spatial model ensembling. This method is derived from the theoretically best approximation of the data generating process (DGP), where the predictions are exact minimax. Two Markov chain Monte Carlo (MCMC) based algorithms are implemented in the BSPS framework for full uncertainty quantification, along with a variational Bayes strategy for faster point inference. This method is also extended for general responses. The examples in this dissertation include multiple simulation studies and two real world data applications. Through these examples, the performance and predictive power of BSPS is shown against various standard spatial models, ensemble methods, and machine learning methods. BSPS is able to maintain predictive accuracy as well as maintain interpretability of the prediction mechanisms. / Statistics

The main limitations in performing uncertainty analysis of CFD models using conventional methods are associated with cost and effort. For these reasons, there is a need for the development and implementation of efficient stochastic CFD tools for performing uncertainty analysis. One of the main contributions of this research is the development and implementation of Intrusive and Non-Intrusive methods using polynomial chaos for uncertainty representation and propagation. In addition, a methodology was developed to address and quantify turbulence model uncertainty. In this methodology, a complex perturbation is applied to the incoming turbulence and closure coefficients of a turbulence model to obtain the sensitivity derivatives, which are used in concert with the polynomial chaos method for uncertainty propagation of the turbulence model outputs. / Ph. D.

The widespread use of high-throughput experimental assays designed to measure the entire complement of a cells genes or gene products has led to vast stores of data which are extremely plentiful in terms of the number of items they can measure in a single sample, yet often sparse in the number of samples per experiment due to their high cost. This often leads to datasets where the number of treatment levels or time points sampled is limited, or where there are very small numbers of technical and/or biological replicates. If the goal is to use this data to infer network models, these sparse datasets can lead to under-determined systems. While model parameter variation and its effects on model robustness has been well studied, most of this work has looked exclusively at accounting for variation only from measurement error. In contrast, little work has been done to isolate and quantify the amount of parameter variation caused by the uncertainty in the unmeasured regions of time course experiments. Here we introduce a novel algorithm to quantify the uncertainty in the unmeasured inter- vals between biological measurements taken across a set of quantitative treatments. The algorithm provides a probabilistic distribution of possible gene expression values within un- measured intervals, based on a plausible biological constraint. We show how quantification of this uncertainty can be used to guide researchers in further data collection by identifying which samples would likely add the most information to the system under study. We also present an application of this method to isolate and quantify two distinct sources of model parameter variation. In the concluding chapter we discuss another source of uncertainty in systems biology, namely gene function prediction, and compare several algorithms designed for that purpose. / Ph. D.

Real world decision-making requires simultaneously determining what we are observing in our environment (perceptual decision-making; PDM) and what the stimuli and actions are worth (reward-based decision-making; RDM). There is evidence of a bi-directional relationship between reward and perceptual information in guiding choice, with some studies suggesting that individuals optimally combine the two. Uncertainty in both reward expectations and perception have been shown to alter choice behavior, however few studies have manipulated both variables simultaneously. Given the distinct theoretical and computational foundations of PDM and RDM, it has also been assumed that the underlying behavioral and neural substrates of perceptual and reward-based choice are separable. However, there is evidence that task structure and subjective value/uncertainty more generally contribute to activity in large-scale networks of the brain, rather than domain specific features (perceptual salience/reward). Variability in task structures and methods of manipulating and modeling sensory and reward uncertainty, make it hard to draw definitive conclusions across these perspectives with currently available data. The current study used behavioral and fMRI techniques to investigate the neurobehavioral substrates of decision-making under simultaneous perceptual and reward uncertainty in a sample of healthy adult volunteers. The primary objectives of this project were to test: a) how simultaneous manipulations in sensory and reward uncertainty influence choice, b) whether task structure alters the influence of sensory and reward information on choice behavior, and c) whether activity in underlying neural substrates reflect domain-specific or domain-general processes. Results showed that choices were best predicted by a combined model of perceptual salience and reward, with an overall bias towards perceptual salience information. Choice percentage was not impacted by task structure, however choices were better predicted by individual features (salience and reward) when they were manipulated stably, than dynamically. Activity in the brain showed greater overlap between dynamic task conditions when compared to both salience and reward conditions. There was also greater overlap between stable task conditions when compared to reward but not salience conditions. Preliminary evidence suggests that activity in decision-relevant regions of the brain varied by uncertainty and value rather than salience and reward per se. / Doctor of Philosophy / Real world decision-making requires knowing what things in our environment are and knowing what they are worth. Uncertainty about what different options are, can affect the actions we take and the outcomes we expect. Uncertainty about the outcome of our choices or actions can also influence how attend to or consider certain options when making choices. We also know that context can affect our behavior. For example, when our environment changes frequently, choices that might have been good in the past, may not necessarily be the best course of action in the future. This can add further confusion about what to do. Across several mental health conditions, we see that problems arise when individuals need to take actions based on incomplete, uncertain, or frequently changing information. Real world decision-making requires knowing what things in our environment are and knowing what they are worth. Uncertainty about what different options are, can affect the actions we take and the outcomes we expect. Uncertainty about the outcome of our choices or actions can also influence how we attend to or consider options when making choices. We also know that context can affect our behavior. For example, when our environment changes frequently, choices that might have been good in the past, may not necessarily be the best course of action in the future. This can add further confusion about what to do. Across several mental health conditions, we see that problems arise when individuals need to take actions based on incomplete, uncertain, or frequently changing information. The first goal of this study was to better understand what healthy individuals do when they are faced with different levels of uncertainty around about what different options are (through changes in visual clarity), and what they are worth (through changes in probability of reward). A second goal was to see whether how the frequency at which choice clarity and outcomes change, effects the kinds of choices people make. Third, the study used a measure of brain activity, to determine what the brain is doing which participants make these complex decisions. Results showed that people's choices were best predicted by considering both clarity of the options and their outcome. Having certainty about the identity of the choices was more important than the value of those choices. Also, information about clarity and value of options were more likely to be considered when they were stable, versus when they were changing frequently. Decision-relevant regions of the brain seemed to respond most to overall information about uncertainty and stability of options rather than their clarity or outcome value per se. Future research should test these findings in a larger sample, further explore individual differences in how people respond to various types of uncertainty and determine how knowledge of these individual differences can inform personalized treatment for individuals with related mental health challenges.

Relative levels of task uncertainty associated with various CPA firm services were examined in this study. Additionally, tests to determine whether systematic variation occurs at the office or at the firm level were conducted. Multiple measures of task uncertainty were developed. Multiple analysis of variance techniques were used to analyze data drawn from audit, tax, actuarial and benefits consulting, and general business consulting engagements. Data was drawn from two office of one Big Six CPA firm. As expected, after comparing audit and tax engagements from two office, the null hypothesis that there was no difference in task uncertainty levels between offices on either service type could not be rejected. The null hypothesis that no difference in levels of task uncertainty between the four service types exist was rejected. This result was also consistent with expectations. These findings provide empirical support for an assumption made by previous researchers that the individual firm is the appropriate level for analysis. Additionally, results suggest that, at the firm level, differences in levels of task uncertainty do exist. In general, audit and tax services appear to involve lower levels of task uncertainty than do consulting services; however, it should be noted that significant differences also existed between consulting services. The implications of these results for future research are that the firm appears to be the appropriate organizational level for examining research questions related to CPA firms. Also, consulting services need to be considered not as one service type, but potentially as distinct from one another. Future research involving other Big six firms as well as second and third-tier firms could lead to greater generalizability of these results / Ph. D.