<p><br></p> <p>Police use of force has become a topic of significant concern, particularly given the disparate impact on communities of color. Research has shown that police officer involved shootings, misconduct and excessive use of force complaints exhibit network effects, where officers are at greater risk of being involved in these incidents when they socialize with officers who have a history of use of force and misconduct. Given that use of force and misconduct behavior appear to be transmissible across police networks, we are attempting to address if police networks can be altered to reduce use of force and misconduct events in a limited scope.</p> <p><br></p> <p>In this work, we analyze a novel dataset from the Indianapolis Metropolitan Police Department on officer field training, subsequent use of force, and the role of network effects from field training officers. We construct a network survival model for analyzing time-to-event of use of force incidents involving new police trainees. The model includes network effects of the diffusion of risk from field training officers (FTOs) to trainees. We then introduce a network rewiring algorithm to maximize the expected time to use of force events upon completion of field training. We study several versions of the algorithm, including constraints that encourage demographic diversity of FTOs. The results show that FTO use of force history is the best predictor of trainee's time to use of force in the survival model and rewiring the network can increase the expected time (in days) of a recruit's first use of force incident by 8%. </p> <p>We then discuss the potential benefits and challenges associated with implementing such an algorithm in practice.</p> <p><br></p>

<p> The hydrological cycle is affected by land use variability. Land use spatial and temporal variability has the power to alter watershed runoff, water resource quantity and quality, ecosystems, and environmental sustainability. In recent decades, agriculture lands, pastures, plantations, and urban areas have increased, resulting in significant increases in energy, water, and fertilizer usage, as well as significant biodiversity losses. </p>

<p>The computer vision task of classifying natural images is a primary driving force behind modern AI algorithms. Deep Convolutional Neural Networks (CNNs) demonstrate state of the art performance in large scale multi-class image classification tasks. However, due to the many layers and millions of parameters these models are considered to be black box algorithms. The decisions of these models are further obscured due to a cumbersome multi-class decision process. There exists another approach called class binarization in the literature which determines the multi-class prediction outcome through a sequence of binary decisions.The focus of this dissertation is on the integration of the class-binarization approach to multi-class classification with deep learning models, such as CNNs, for addressing large scale image classification problems. Three works are presented to address the integration.</p> <p>In the first work, Error Correcting Output Codes (ECOCs) are integrated into CNNs by inserting a latent-binarization layer prior to the CNNs final classification layer.  This approach encapsulates both encoding and decoding steps of ECOC into a single CNN architecture. EM and Gibbs sampling algorithms are combined with back-propagation to train CNN models with Latent Binarization (LB-CNN). The training process of LB-CNN guides the model to discover hidden relationships similar to the semantic relationships known apriori between the categories. The proposed models and algorithms are applied to several image recognition tasks, producing excellent results.</p> <p>In the second work, Hierarchically Decodeable Output Codes (HD-OCs) are proposedto compactly describe a hierarchical probabilistic binary decision process model over the features of a CNN. HD-OCs enforce more homogeneous assignments of the categories to the dichotomy labels. A novel concept called average decision depth is presented to quantify the average number of binary questions needed to classify an input. An HD-OC is trained using a hierarchical log-likelihood loss that is empirically shown to orient the output of the latent feature space to resemble the hierarchical structure described by the HD-OC. Experiments are conducted at several different scales of category labels. The experiments demonstrate strong performance and powerful insights into the decision process of the model.</p> <p>In the final work, the literature of enumerative combinatorics and partially ordered sets isused to establish a unifying framework of class-binarization methods under the Multivariate Bernoulli family of models. The unifying framework theoretically establishes simple relationships for transitioning between the different binarization approaches. Such relationships provide useful investigative tools for the discovery of statistical dependencies between large groups of categories. They are additionally useful for incorporating taxonomic information as well as enforcing structural model constraints. The unifying framework lays the groundwork for future theoretical and methodological work in addressing the fundamental issues of large scale multi-class classification.</p> <p><br></p>

<p>This dissertation address two significant challenges in the causal inference workflow for Big Observational Data. The first is designing Big Observational Data with high-dimensional and heterogeneous covariates. The second is performing uncertainty quantification for estimates of causal estimands that are obtained from the application of black box machine learning algorithms on the designed Big Observational Data. The methodologies developed by addressing these challenges are applied for the design and analysis of Big Observational Data from a large public university in the United States. </p> <h4>Distributed Design</h4> <p>A fundamental issue in causal inference for Big Observational Data is confounding due to covariate imbalances between treatment groups. This can be addressed by designing the study prior to analysis. The design ensures that subjects in the different treatment groups that have comparable covariates are subclassified or matched together. Analyzing such a designed study helps to reduce biases arising from the confounding of covariates with treatment. Existing design methods, developed for traditional observational studies consisting of a single designer, can yield unsatisfactory designs with sub-optimum covariate balance for Big Observational Data due to their inability to accommodate the massive dimensionality, heterogeneity, and volume of the Big Data. We propose a new framework for the distributed design of Big Observational Data amongst collaborative designers. Our framework first assigns subsets of the high-dimensional and heterogeneous covariates to multiple designers. The designers then summarize their covariates into lower-dimensional quantities, share their summaries with the others, and design the study in parallel based on their assigned covariates and the summaries they receive. The final design is selected by comparing balance measures for all covariates across the candidates and identifying the best amongst the candidates. We perform simulation studies and analyze datasets from the 2016 Atlantic Causal Inference Conference Data Challenge to demonstrate the flexibility and power of our framework for constructing designs with good covariate balance from Big Observational Data.</p> <h4>Designed Bootstrap</h4> <p>The combination of modern machine learning algorithms with the nonparametric bootstrap can enable effective predictions and inferences on Big Observational Data. An increasingly prominent and critical objective in such analyses is to draw causal inferences from the Big Observational Data. A fundamental step in addressing this objective is to design the observational study prior to the application of machine learning algorithms. However, the application of the traditional nonparametric bootstrap on Big Observational Data requires excessive computational efforts. This is because every bootstrap sample would need to be re-designed under the traditional approach, which can be prohibitive in practice. We propose a design-based bootstrap for deriving causal inferences with reduced bias from the application of machine learning algorithms on Big Observational Data. Our bootstrap procedure operates by resampling from the original designed observational study. It eliminates the need for additional, costly design steps on each bootstrap sample that are performed under the standard nonparametric bootstrap. We demonstrate the computational efficiency of this procedure compared to the traditional nonparametric bootstrap, and its equivalency in terms of confidence interval coverage rates for the average treatment effects, by means of simulation studies and a real-life case study.</p> <h4>Case Study</h4> <p>We apply the distributed design and designed bootstrap methodologies in a case study involving institutional data from a large public university in the United States. The institutional data contains comprehensive information about the undergraduate students in the university, ranging from their academic records to on-campus activities. We study the causal effects of undergraduate students’ attempted course load on their academic performance based on a selection of covariates from these data. Ultimately, our real-life case study demonstrates how our methodologies enable researchers to effectively use straightforward design procedures to obtain valid causal inferences with reduced computational efforts from the application of machine learning algorithms on Big Observational Data.</p> <p><br></p>

<p dir="ltr">We propose a fully online procedure to conduct statistical inference with adaptively collected data. The low-rank structure of the model parameter and the adaptivity nature of the data collection process make this task challenging: standard low-rank estimators are biased and cannot be obtained in a sequential manner while existing inference approaches in sequential decision-making algorithms fail to account for the low-rankness and are also biased. To tackle the challenges previously outlined, we first develop an online low-rank estimation process employing Stochastic Gradient Descent with noisy observations. Subsequently, to facilitate statistical inference using the online low-rank estimator, we introduced a novel online debiasing technique designed to address both sources of bias simultaneously. This method yields an unbiased estimator suitable for parameter inference. Finally, we developed an inferential framework capable of establishing an online estimator for performing inference on the optimal policy value. In theory, we establish the asymptotic normality of the proposed online debiased estimators and prove the validity of the constructed confidence intervals for both inference tasks. Our inference results are built upon a newly developed low-rank stochastic gradient descent estimator and its non-asymptotic convergence result, which is also of independent interest.</p>

<p dir="ltr">Statistical arbitrage is a prevalent trading strategy which takes advantage of mean reverse property of spreads constructed from pairs or portfolios of assets. Utilizing statistical models and algorithms, statistical arbitrage exploits and capitalizes on the pricing inefficiencies between securities or within asset portfolios. </p><p dir="ltr">In chapter 2, We propose a framework for constructing diversified portfolios with multiple pairs trading strategies. In our approach, several pairs of co-moving assets are traded simultaneously, and capital is dynamically allocated among different pairs based on the statistical characteristics of the historical spreads. This allows us to further consider various portfolio designs and rebalancing strategies. Working with empirical data, our experiments suggest the significant benefits of diversification within our proposed framework.</p><p dir="ltr">In chapter 3, we explore an optimal timing strategy for the trading of price spreads exhibiting mean-reverting characteristics. A sequential optimal stopping framework is formulated to analyze the optimal timings for both entering and subsequently liquidating positions, all while considering the impact of transaction costs. Then we leverages a refined signature optimal stopping method to resolve this sequential optimal stopping problem, thereby unveiling the precise entry and exit timings that maximize gains. Our framework operates without any predefined assumptions regarding the dynamics of the underlying mean-reverting spreads, offering adaptability to diverse scenarios. Numerical results are provided to demonstrate its superior performance when comparing with conventional mean reversion trading rules.</p><p dir="ltr">In chapter 4, we introduce an innovative model-free and reinforcement learning based framework for statistical arbitrage. For the construction of mean reversion spreads, we establish an empirical reversion time metric and optimize asset coefficients by minimizing this empirical mean reversion time. In the trading phase, we employ a reinforcement learning framework to identify the optimal mean reversion strategy. Diverging from traditional mean reversion strategies that primarily focus on price deviations from a long-term mean, our methodology creatively constructs the state space to encapsulate the recent trends in price movements. Additionally, the reward function is carefully tailored to reflect the unique characteristics of mean reversion trading.</p>

<p>Graphs are a natural choice for understanding the associations between variables, and assuming a probabilistic embedding for the graph structure leads to a variety of graphical models that enable us to understand these associations even further. In the realm of high-dimensional data, where the number of associations between interacting variables is far greater than the available number of data points, the goal is to infer a sparse graph. In this thesis, we make contributions in the domain of Bayesian graphical models, where our prior belief on the graph structure, encoded via uncertainty on the model parameters, enables the estimation of sparse graphs.</p> <p><br></p> <p>We begin with the Gaussian Graphical Model (GGM) in Chapter 2, one of the simplest and most famous graphical models, where the joint distribution of interacting variables is assumed to be Gaussian. In GGMs, the conditional independence among variables is encoded in the inverse of the covariance matrix, also known as the precision matrix. Under a Bayesian framework, we propose a novel prior--penalty dual called the `graphical horseshoe-like' prior and penalty, to estimate precision matrix. We also establish the posterior convergence of the precision matrix estimate and the frequentist consistency of the maximum a posteriori (MAP) estimator.</p> <p><br></p> <p>In Chapter 3, we develop a general framework based on local linear approximation for MAP estimation of the precision matrix in GGMs. This general framework holds true for any graphical prior, where the element-wise priors can be written as a Laplace scale mixture. As an application of the framework, we perform MAP estimation of the precision matrix under the graphical horseshoe penalty.</p> <p><br></p> <p>In Chapter 4, we focus on graphical models where the joint distribution of interacting variables cannot be assumed Gaussian. Motivated by the quantile graphical models, where the Gaussian likelihood assumption is relaxed, we draw inspiration from the domain of precision medicine, where personalized inference is crucial to tailor individual-specific treatment plans. With an aim to infer Directed Acyclic Graphs (DAGs), we propose a novel quantile DAG learning framework, where the DAGs depend on individual-specific covariates, making personalized inference possible. We demonstrate the potential of this framework in the regime of precision medicine by applying it to infer protein-protein interaction networks in Lung adenocarcinoma and Lung squamous cell carcinoma.</p> <p><br></p> <p>Finally, we conclude this thesis in Chapter 5, by developing a novel framework to compute the marginal likelihood in a GGM, addressing a longstanding open problem. Under this framework, we can compute the marginal likelihood for a broad class of priors on the precision matrix, where the element-wise priors on the diagonal entries can be written as gamma or scale mixtures of gamma random variables and those on the off-diagonal terms can be represented as normal or scale mixtures of normal. This result paves new roads for model selection using Bayes factors and tuning of prior hyper-parameters.</p>

<p dir="ltr">The primary purpose of this thesis is to propose and demonstrate BioGRNsemble, a modular and flexible approach for inferencing gene regulatory networks from RNA-Seq data. Integrating the GENIE3 and GRNBoost2 algorithms, this ensembles-of-ensembles method attempts to balance the outputs of both models through averaging, before providing a trimmed-down gene regulatory network consisting of transcription and target genes. Using a Drosophila Eye Dataset, we were able to successfully test this novel methodology, and our validation analysis using an online database determined over 3500 gene links correctly detected, albeit out of almost 530,000 predictions, leaving plenty of room for improvement in the future.</p>

<p dir="ltr">The generalization error of a classifier is related to the complexity of the set of functions among which the classifier is chosen. We study a family of low-complexity classifiers consisting of thresholding a random one-dimensional feature. The feature is obtained by projecting the data on a random line after embedding it into a higher-dimensional space parametrized by monomials of order up to k. More specifically, the extended data is projected n-times and the best classifier among those n, based on its performance on training data, is chosen. </p><p dir="ltr">We show that this type of classifier is extremely flexible, as it is likely to approximate, to an arbitrary precision, any continuous function on a compact set as well as any Boolean function on a compact set that splits the support into measurable subsets. In particular, given full knowledge of the class conditional densities, the error of these low-complexity classifiers would converge to the optimal (Bayes) error as k and n go to infinity. On the other hand, if only a training dataset is given, we show that the classifiers will perfectly classify all the training points as k and n go to infinity. </p><p dir="ltr">We also bound the generalization error of our random classifiers. In general, our bounds are better than those for any classifier with VC dimension greater than O(ln(n)). In particular, our bounds imply that, unless the number of projections n is extremely large, there is a significant advantageous gap between the generalization error of the random projection approach and that of a linear classifier in the extended space. Asymptotically, as the number of samples approaches infinity, the gap persists for any such n. Thus, there is a potentially large gain in generalization properties by selecting parameters at random, rather than optimization. </p><p dir="ltr">Given a classification problem and a family of classifiers, the Rashomon ratio measures the proportion of classifiers that yield less than a given loss. Previous work has explored the advantage of a large Rashomon ratio in the case of a finite family of classifiers. Here we consider the more general case of an infinite family. We show that a large Rashomon ratio guarantees that choosing the classifier with the best empirical accuracy among a random subset of the family, which is likely to improve generalizability, will not increase the empirical loss too much. </p><p dir="ltr">We quantify the Rashomon ratio in two examples involving infinite classifier families in order to illustrate situations in which it is large. In the first example, we estimate the Rashomon ratio of the classification of normally distributed classes using an affine classifier. In the second, we obtain a lower bound for the Rashomon ratio of a classification problem with a modified Gram matrix when the classifier family consists of two-layer ReLU neural networks. In general, we show that the Rashomon ratio can be estimated using a training dataset along with random samples from the classifier family and we provide guarantees that such an estimation is close to the true value of the Rashomon ratio.</p>