What is the probability that in a fair coin toss game (a simple random walk) we go bankrupt in n steps when there is an initial lead of some known or unknown quantity $m? What is the distribution of the number of steps N that it takes for the lead to vanish? This thesis explores some of the features of this first passage to the origin (FPO) distribution. First, we explore the distribution of N when m is known. Next, we compute the maximum likelihood estimators of m for a fixed n and also the posterior distribution of m when we are given that m follows some known prior distribution.

This dissertation consists of three chapters exploring education and crime in the modern economy. The first two chapters focus on inter-district school choice and teacher labor markets in Massachusetts. The third chapter examines the demolition of public housing in Chicago and its interaction with the geospatial distribution of gang territory. In the first chapter, I study the sorting of students to school districts using new lottery data from an inter-district school choice program. I find that moving to a more preferred school district generates benefits to student test scores, coursework quality, high-school graduation, and college attendance. Motivated by these findings, I develop a rich model of treatment effect heterogeneity and estimate it using a new empirical-Bayes-type procedure that leverages non-experimental data to increase precision in quasi-experimental designs. I use the heterogeneous effects to show that nearly all the test score gains from the choice program emerge from Roy selection. In the second chapter (joint with Scott Imberman and Marcus Winters), we describe the relationship between school quality, teacher value-added, and teacher attrition across the public and charter sectors. We begin by documenting important differences in the sources of variation that explain attrition across sectors. Next we demonstrate that while charters are in fact more likely to remove their worst teachers, they are also more likely to lose their best. We conclude by exploring the type and quality of destination schools among teachers who move. In the third chapter, I study the demolition of 22,000 units of public housing on crime in Chicago. Point estimates that incorporate both the direct and spillover effects indicate that in the short run, the average demolition increased city-wide crime by 0.5% per month relative to baseline, with no evidence of offsetting long run reductions. I also provide evidence that spillovers are mediated by demolition-induced migration across gang territorial boundaries. I reconcile my findings with contradictory results from the existing literature by proposing and applying a test for control group contamination. I find that existing results are likely biased by previously unaccounted for spillovers.

Structural equation modelling (SEM) is carried out with the aim of testing hypotheses on the model of the researcher in a quantitative way, using the sampled data. Although SEM has developed in many aspects over the past few decades, there are still numerous advances which can make SEM an even more powerful technique. We propose representing the nal theoretical SEM by a Bayesian Network (BN), which we would like to call a Probabilistic Structural Equation Model (PSEM). With the PSEM, we can take things a step further and conduct inference by explicitly entering evidence into the network and performing di erent types of inferences. Because the direction of the inference is not an issue, various scenarios can be simulated using the BN. The augmentation of SEM with BN provides signi cant contributions to the eld. Firstly, structural learning can mine data for additional causal information which is not necessarily clear when hypothesising causality from theory. Secondly, the inference ability of the BN provides not only insight as mentioned before, but acts as an interactive tool as the `what-if' analysis is dynamic. / Mini Dissertation (MCom)--University of Pretoria, 2018. / Statistics / MCom / Unrestricted

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The need to cope with the rapid transformation of the conventional electrical grid into the future smart grid, with multiple connected microgrids, has led to the investigation of optimal smart grid architectures. The main components of the future smart grids such as generators, substations, controllers, smart meters and collector nodes are evolving; however, truly effective integration of these elements into the microgrid context to guarantee intelligent and dynamic functionality across the whole smart grid remains an open issue. Energy trading is a significant part of this integration. In microgrids, energy trading refers to the use of surplus energy in one microgrid to satisfy the demand of another microgrid or a group of microgrids that form a microgrid community. Different techniques are employed to manage the energy trading process such as optimization-based and conventional game-theoretical methods, which bring about several challenges including complexity, scalability and ability to learn dynamic environments. A common challenge among all of these methods is adapting to changing circumstances. Optimization methods, for example, show promising performance in static scenarios where the optimal solution is achieved for a specific snapshot of the system. However, to use such a technique in a dynamic environment, finding the optimal solutions for all the time slots is needed, which imposes a significant complexity. Challenges such as this can be best addressed using game theory techniques empowered with machine learning methods across grid infrastructure and microgrid communities. In this thesis, novel Bayesian coalitional game theory-based and Bayesian reinforcement learning-based coalition formation algorithms are proposed, which allow the microgrids to exchange energy with their coalition members while minimizing the associated cost and power loss. In addition, a deep reinforcement learning scheme is developed to address the problem of large convergence time resulting from the sizeable state-action space of the methods mentioned above. The proposed algorithms can ideally overcome the uncertainty in the system. The advantages of the proposed methods are highlighted by comparing them with the conventional coalitional game theory-based techniques, Q-learning-based technique, random coalition formation, as well as with the case with no coalitions. The results show the superiority of the proposed methods in terms of power loss and cost minimization in dynamic environments.

In 2015, Illinois changed size and harvest limits for catfishes (blue catfish Ictalurus furcatus, flathead catfish Pylodictis olivaris, and channel catfish Ictalurus punctatus) in the Ohio River to match those of neighboring states in order to provide continuity of the regulations and promote a trophy catfish fishery. Regulations imposed a daily limit of one blue catfish or flathead catfish ≥ 35 inches (88.9 cm) and one channel catfish ≥ 28 inches (71.1 cm) per fisher and a 13 inch (33.0 cm) minimum length limit for all species with no bag limit. Although management regulations were implemented, potential efficacy of the implemented regulations and appropriate (i.e. most precise or accurate with fewest samples) monitoring protocols were unknown. Furthermore, there was general lack of understanding of early life movements, natal dispersal timing and principal recruitment sources that aide in determining appropriate spatial scale for monitoring and managing lower Ohio River catfish stocks.To fill these knowledge gaps the following methods were employed: 1) simulation modeling was used to evaluate precision in estimating catch and size distribution metrics for monitoring population trends with increasing sample size (i.e., sampling events), 2) N-mixture modeling was used to estimate size selectivity of multiple gears using detection probability as a robust alternative to size-specific catchability coefficients, 3) otolith microchemistry (Sr:Ca and Ba:Ca) was employed to determine principal recruitment sources, early life movement patterns, and provide fisheries managers with a better understanding of the spatial extent to which management actions should be implemented, 4) Bayesian modeling was used to estimate growth and mortality, 5) Yield-per-recruit modeling was used to estimate and evaluate fishing mortality rates that would result in growth overfishing (FMAX) and yield at FMAX (YPRMAX) for three management scenarios (no regulation, minimum length limit [33.0 cm or greater] and a permissive slot limit [33.0 cm – 88.9 cm; blue catfish and flathead catfish]). The simulation models presented account for the uncertainty associated with heterogeneous selectivity of a gear, and minimize the impact of rare or extreme catch values. Trotlines and low pulse (15-pps) electrofishing generally required the fewer samples to achieve stable values of catch per unit of effort (CPUE), proportional size distribution (quality; PSDQ), and coefficient of variation (CV) than other gears based on simulation modeling. Abundance and detection probabilities were estimated separately for each species of catfish by length category within and across gears, producing a species-gear-size correction for catch bias used in estimating Proportional Size Distribution - Quality (PSDQ). Corrected (i.e., accounting for detection) PSDQ values were lower than uncorrected estimates suggesting a positive bias for larger fish across the entire sampling regime. Managers should use a combination of low pulse electrofishing, trotlines, and high pulse (60-pps) electrofishing in their monitoring efforts for all three species. Based on microchemistry, ictalurid catfishes in the lower Ohio River appear to recruit from multiple sources and make movements across a broad geographic scale. Additionally, some catfish may be originating from outside the portion of the Ohio River that is managed by Illinois (lower 214 km). Fisheries managers should take this into account when implementing management actions. However, most ictalurid catfishes originated from riverine (e.g., Ohio and Mississippi River) natal environments and not from smaller tributaries, and managers should not expect tributaries to compensate for weak year-classes within the river. Based on yield per recruit modeling, catfish stocks are unlikely to benefit from current regulations or a theoretical minimum size limit given the near complete overlap of YPRMAX confidence intervals for all estimable scenarios and the small statistical difference (1 – 3%) based on FMAX between the most permissive and most restrictive scenarios. While statistical differences in FMAX exist, they are likely biologically irrelevant, exceeding the precision of estimation methods for F. While there is some indication that alignment and continuity of management regulations is warranted based on microchemistry, the efficacy of the current permissive slot regulations is questionable based on the models presented and the life history of these fishes. There is no advantage to implementing any of the modeled regulations in terms of increasing FMAX or YPRMAX.

In this thesis, I explore in two essays how to augment thin historical purchase data with other sources of information using Bayesian and probabilistic machine learning frameworks to better infer customers' preferences and their future behavior. In the first essay, I posit that firms can better manage recently-acquired customers by using the information from acquisition to inform future demand preferences for those customers. I develop a probabilistic machine learning model based on Deep Exponential Families to relate multiple acquisition characteristics with individual level demand parameters, and I show that the model is able to capture flexibly non-linear relationships between acquisition behaviors and demand parameters. I estimate the model using data from a retail context and show that firms can better identify which new customers are the most valuable. In the second essay, I explore how to combine the information collected through the customer journey—search queries, clicks and purchases; both within-journeys and across journeys—to infer the customer’s preferences and likelihood of buying, in settings in which there is thin purchase history and where preferences might change from one purchase journey to another. I propose a non-parametric Bayesian model that combines these different sources of information and accounts for what I call context heterogeneity, which are journey-specific preferences that depend on the context of the specific journey. I apply the model in the context of airline ticket purchases using data from one of the largest travel search websites and show that the model is able to accurately infer preferences and predict choice in an environment characterized by very thin historical data. I find strong context heterogeneity across journeys, reinforcing the idea that treating all journeys as stemming from the same set of preferences may lead to erroneous inferences.

Innovations in neural data recording techniques are revolutionizing neuroscience and presenting both challenges and opportunities for statistical data analysis. This dissertation discusses several recent advances in neural data signal processing, encoding, decoding, and dimension reduction. Chapter 1 introduces challenges in neural data science and common statistical methods used to address them. Chapter 2 develops a new method to detect neurons and extract signals from noisy calcium imaging data with irregular neuron shapes. Chapter 3 introduces a novel probabilistic framework for modeling deconvolved calcium traces. Chapter 4 proposes an improved Bayesian nonparametric extension of the hidden Markov model (HMM) that separates the strength of the self-persistence prior and transition prior. Chapter 5 introduces a more identifiable and interpretable latent variable model for Poisson observations. We develop efficient algorithms to fit each of the aforementioned methods and demonstrate their effectiveness on both simulated and real data.

In light of advancements made in data collection techniques over the past two decades, data mining has become common practice to summarize large, high dimensional datasets, in hopes of discovering noteworthy data structures. However, one concern is that most data mining approaches rely upon strict criteria that may mask information in data that analysts may find useful. We propose a new approach called Bayesian Visual Analytics (BaVA) which merges Bayesian Statistics with Visual Analytics to address this concern. The BaVA framework enables experts to interact with the data and the feature discovery tools by modeling the "sense-making" process using Bayesian Sequential Updating. In this paper, we use BaVA idea to enhance high dimensional visualization techniques such as Probabilistic PCA (PPCA). However, for real-world datasets, important structures can be arbitrarily complex and a single data projection such as PPCA technique may fail to provide useful insights. One way for visualizing such a dataset is to characterize it by a mixture of local models. For example, Tipping and Bishop [Tipping and Bishop, 1999] developed an algorithm called Mixture Probabilistic PCA (MPPCA) that extends PCA to visualize data via a mixture of projectors. Based on MPPCA, we developped a new visualization algorithm called Covariance-Guided MPPCA which group similar covariance structured clusters together to provide more meaningful and cleaner visualizations. Another way to visualize a very complex dataset is using nonlinear projection methods such as  the Generative Topographic Mapping algorithm(GTM). We developped an interactive version of GTM to discover interesting local data structures. We demonstrate the performance of our approaches using both synthetic and real dataset and compare our algorithms with existing ones. / Ph. D.

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