Given data on the choices made by consumers for different offer sets, a key challenge is to develop parsimonious models that describe and predict consumer choice behavior while being amenable to prescriptive tasks such as pricing and assortment optimization. The marginal distribution model (MDM) is one such model, that requires only the specification of marginal distributions of the random utilities. This paper aims to establish necessary and sufficient conditions for given choice data to be consistent with the MDM hypothesis, inspired by the utility of similar characterizations for the random utility model (RUM). This endeavor leads to an exact characterization of the set of choice probabilities that the MDM can represent. Verifying the consistency of choice data with this characterization is equivalent to solving a polynomial-sized linear program. Since the analogous verification task for RUM is computationally intractable and neither of these models subsumes the other, MDM is helpful in striking a balance between tractability and representational power. The characterization is convenient to be used with robust optimization for making data-driven sales and revenue predictions for new unseen assortments. When the choice data lacks consistency with the MDM hypothesis, finding the best-fitting MDM choice probabilities reduces to solving a mixed integer convex program. The results extend naturally to the case where the alternatives can be grouped based on the similarity of the marginal distributions of the utilities. Numerical experiments show that MDM provides better representational power and prediction accuracy than multinominal logit and significantly better computational performance than RUM.

The availability of both structured and unstructured databases, such as electronic health data, social media data, patent data, and surveys that are often updated in real time, among others, has grown rapidly over the past decade. With this expansion, the statistical and methodological questions around data integration, or rather merging multiple data sources, has also grown. Specifically, the science of the "data cleaning pipeline" contains four stages that allow an analyst to perform downstream tasks, predictive analyses, or statistical analyses on "cleaned data." This article provides a review of this emerging field, introducing technical terminology and commonly used methods. Statement of Significance: The article reviews the data cleaning pipeline, introducing technical terminology and commonly used methods.

Autonomous robots have real-world applications in diverse fields, such as mobile manipulation and environmental exploration, and many such tasks benefit from a hands-off approach in terms of human user involvement over a long task horizon. However, the level of autonomy achievable by a deployment is limited in part by the problem definition or task specification required by the system. Task specifications often require technical, low-level information that is unintuitive to describe and may result in generic solutions, burdening the user technically both before and after task completion. In this thesis, we aim to advance task specification abstraction toward the goal of increasing robot autonomy in real-world scenarios. We do so by tackling problems that address several different angles of this goal. First, we develop a way for the automatic discovery of optimal transition points between subtasks in the context of constrained mobile manipulation, removing the need for the human to hand-specify these in the task specification. We further propose a way to automatically describe constraints on robot motion by using demonstrated data as opposed to manually-defined constraints. Then, within the context of environmental exploration, we propose a flexible task specification framework, requiring just a set of quantiles of interest from the user that allows the robot to directly suggest locations in the environment for the user to study. We next systematically study the effect of including a robot team in the task specification and show that multirobot teams have the ability to improve performance under certain specification conditions, including enabling inter-robot communication. Finally, we propose methods for a communication protocol that autonomously selects useful but limited information to share with the other robots.

We consider the problem of selecting a small subset of representative variables from a large dataset. In the computer science literature, this dimensionality reduction problem is typically formalized as Column Subset Selection (CSS). Meanwhile, the typical statistical formalization is to find an information-maximizing set of Principal Variables. This paper shows that these two approaches are equivalent, and moreover, both can be viewed as maximum likelihood estimation within a certain semi-parametric model. Using these connections, we show how to efficiently (1) perform CSS using only summary statistics from the original dataset; (2) perform CSS in the presence of missing and/or censored data; and (3) select the subset size for CSS in a hypothesis testing framework.

We introduce a multifidelity estimator of covariance matrices formulated as the solution to a regression problem on the manifold of symmetric positive definite matrices. The estimator is positive definite by construction, and the Mahalanobis distance minimized to obtain it possesses properties which enable practical computation. We show that our manifold regression multifidelity (MRMF) covariance estimator is a maximum likelihood estimator under a certain error model on manifold tangent space. More broadly, we show that our Riemannian regression framework encompasses existing multifidelity covariance estimators constructed from control variates. We demonstrate via numerical examples that our estimator can provide significant decreases, up to one order of magnitude, in squared estimation error relative to both single-fidelity and other multifidelity covariance estimators. Furthermore, preservation of positive definiteness ensures that our estimator is compatible with downstream tasks, such as data assimilation and metric learning, in which this property is essential.

We suggest a simple Gaussian mixture model for data generation that complies with Feldman's long tail theory (2020). We demonstrate that a linear classifier cannot decrease the generalization error below a certain level in the proposed model, whereas a nonlinear classifier with a memorization capacity can. This confirms that for long-tailed distributions, rare training examples must be considered for optimal generalization to new data. Finally, we show that the performance gap between linear and nonlinear models can be lessened as the tail becomes shorter in the subpopulation frequency distribution, as confirmed by experiments on synthetic and real data.

Reinforcement learning (RL) has been shown to learn sophisticated control policies for complex tasks including games, robotics, heating and cooling systems and text generation. The action-perception cycle in RL, however, generally assumes that a measurement of the state of the environment is available at each time step without a cost. In applications such as materials design, deep-sea and planetary robot exploration and medicine, however, there can be a high cost associated with measuring, or even approximating, the state of the environment. In this paper, we survey the recently growing literature that adopts the perspective that an RL agent might not need, or even want, a costly measurement at each time step. Within this context, we propose the Deep Dynamic Multi-Step Observationless Agent (DMSOA), contrast it with the literature and empirically evaluate it on OpenAI gym and Atari Pong environments. Our results, show that DMSOA learns a better policy with fewer decision steps and measurements than the considered alternative from the literature.

In this work, minibatch MCMC sampling for feedforward neural networks is made more feasible. To this end, it is proposed to sample subgroups of parameters via a blocked Gibbs sampling scheme. By partitioning the parameter space, sampling is possible irrespective of layer width. It is also possible to alleviate vanishing acceptance rates for increasing depth by reducing the proposal variance in deeper layers. Increasing the length of a non-convergent chain increases the predictive accuracy in classification tasks, so avoiding vanishing acceptance rates and consequently enabling longer chain runs have practical benefits. Moreover, non-convergent chain realizations aid in the quantification of predictive uncertainty. An open problem is how to perform minibatch MCMC sampling for feedforward neural networks in the presence of augmented data.

Motivated by the need for the development of safe and reliable methods for uncertainty quantification in machine learning, I propose and develop ideas for a model-free statistical framework for imprecise probabilistic prediction inference. This framework facilitates uncertainty quantification in the form of prediction sets that offer finite sample control of type 1 errors, a property shared with conformal prediction sets, but this new approach also offers more versatile tools for imprecise probabilistic reasoning. Furthermore, I propose and consider the theoretical and empirical properties of a precise probabilistic approximation to the model-free imprecise framework. Approximating a belief/plausibility measure pair by an [optimal in some sense] probability measure in the credal set is a critical resolution needed for the broader adoption of imprecise probabilistic approaches to inference in statistical and machine learning communities. It is largely undetermined in the statistical and machine learning literatures, more generally, how to properly quantify uncertainty in that there is no generally accepted standard of accountability of stated uncertainties. The research I present in this manuscript is aimed at motivating a framework for statistical inference with reliability and accountability as the guiding principles.

Prognostic and Health Management (PHM) has been widely applied to hardware systems in the electronics and non-electronics domains but has not been explored for software. While software does not decay over time, it can degrade over release cycles. Software health management is confined to diagnostic assessments that identify problems, whereas prognostic assessment potentially indicates when in the future a problem will become detrimental. Relevant research areas such as software defect prediction, software reliability prediction, predictive maintenance of software, software degradation, and software performance prediction, exist, but all of these represent diagnostic models built upon historical data, none of which can predict an RUL for software. This paper addresses the application of PHM concepts to software systems for fault predictions and RUL estimation. Specifically, this paper addresses how PHM can be used to make decisions for software systems such as version update and upgrade, module changes, system reengineering, rejuvenation, maintenance scheduling, budgeting, and total abandonment. This paper presents a method to prognostically and continuously predict the RUL of a software system based on usage parameters (e.g., the numbers and categories of releases) and performance parameters (e.g., response time). The model developed has been validated by comparing actual data, with the results that were generated by predictive models. Statistical validation (regression validation, and k-fold cross validation) has also been carried out. A case study, based on publicly available data for the Bugzilla application is presented. This case study demonstrates that PHM concepts can be applied to software systems and RUL can be calculated to make system management decisions.