Spatial data are characterized by their spatial dependence, which is often complex, non-linear, and difficult to capture with a single model. Significant levels of model uncertainty -- arising from these characteristics -- cannot be resolved by model selection or simple ensemble methods. We address this issue by proposing a novel methodology that captures spatially varying model uncertainty, which we call Bayesian spatial predictive synthesis. Our proposal is derived by identifying the theoretically best approximate model under reasonable conditions, which is a latent factor spatially varying coefficient model in the Bayesian predictive synthesis framework. We then show that our proposed method produces exact minimax predictive distributions, providing finite sample guarantees. Two MCMC strategies are implemented for full uncertainty quantification, as well as a variational inference strategy for fast point inference. We also extend the estimation strategy for general responses. Through simulation examples and two real data applications, we demonstrate that our proposed spatial Bayesian predictive synthesis outperforms standard spatial models and advanced machine learning methods in terms of predictive accuracy.

Qualitative probabilistic networks (QPNs) combine the conditional independence assumptions of Bayesian networks with the qualitative properties of positive and negative dependence. They formalise various intuitive properties of positive dependence to allow inferences over a large network of variables. However, we will demonstrate in this paper that, due to an incorrect symmetry property, many inferences obtained in non-binary QPNs are not mathematically true. We will provide examples of such incorrect inferences and briefly discuss possible resolutions.

This dissertation focuses on modern causal inference under uncertainty and data restrictions, with applications to neoadjuvant clinical trials, distributed data networks, and robust individualized decision making. In the first project, we propose a method under the principal stratification framework to identify and estimate the average treatment effects on a binary outcome, conditional on the counterfactual status of a post-treatment intermediate response. Under mild assumptions, the treatment effect of interest can be identified. We extend the approach to address censored outcome data. The proposed method is applied to a neoadjuvant clinical trial and its performance is evaluated via simulation studies. In the second project, we propose a tree-based model averaging approach to improve the estimation accuracy of conditional average treatment effects at a target site by leveraging models derived from other potentially heterogeneous sites, without them sharing subject-level data. The performance of this approach is demonstrated by a study of the causal effects of oxygen therapy on hospital survival rates and backed up by comprehensive simulations. In the third project, we propose a robust individualized decision learning framework with sensitive variables to improve the worst-case outcomes of individuals caused by sensitive variables that are unavailable at the time of decision. Unlike most existing work that uses mean-optimal objectives, we propose a robust learning framework by finding a newly defined quantile- or infimum-optimal decision rule. From a causal perspective, we also generalize the classic notion of (average) fairness to conditional fairness for individual subjects. The reliable performance of the proposed method is demonstrated through synthetic experiments and three real-data applications.

Counterfactual explanations utilize feature perturbations to analyze the outcome of an original decision and recommend an actionable recourse. We argue that it is beneficial to provide several alternative explanations rather than a single point solution and propose a probabilistic paradigm to estimate a diverse set of counterfactuals. Specifically, we treat the perturbations as random variables endowed with prior distribution functions. This allows sampling multiple counterfactuals from the posterior density, with the added benefit of incorporating inductive biases, preserving domain specific constraints and quantifying uncertainty in estimates. More importantly, we leverage Bayesian hierarchical modeling to share information across different subgroups of a population, which can both improve robustness and measure fairness. A gradient based sampler with superior convergence characteristics efficiently computes the posterior samples. Experiments across several datasets demonstrate that the counterfactuals estimated using our approach are valid, sparse, diverse and feasible.

We evaluate the state estimation performance of the RL-ROE for systems governed by the Burgers equation and Navier-Stokes equations. For each system, we first compute various solution trajectories corresponding to different physical parameter values, which we use to construct the ROM and train the RL-ROE following the procedure outlined in Section 2.4. The trained RL-ROE is finally deployed online and compared against a time-dependent Kalman filter constructed from the same ROM, which we refer to as KF-ROE. The KF-ROE is given by equations (3a) and (4), with the calculation of the time-varying Kalman gain detailed in Appendix C of the supplementary materials. Before proceeding to the results, we discuss our choice of baseline. The ensemble Kalman filter and 4D-Var are two estimation techniques for high-dimensional systems such as those governed by PDEs (Lorenc, 2003). Although they are commonly employed for data assimilation in numerical weather prediction, they require large computational resources since they involve repeated solutions of the high-dimensional dynamics (1). Thus, they are not applicable in the context of embedded control systems, whose limited resources call for an inexpensive model such as the ROM (2). Since the ROM that we consider has linear dynamics, extensions of the Kalman filter for nonlinear dynamics such as the extended or unscented Kalman filters (Wan & Van Der Merwe, 2000; Julier & Uhlmann, 2004) are not relevant, and the vanilla Kalman filter remains the best choice of baseline.

This paper considers the problem of system identification (ID) of linear and nonlinear non-autonomous systems from noisy and sparse data. We propose and analyze an objective function derived from a Bayesian formulation for learning a hidden Markov model with stochastic dynamics. We then analyze this objective function in the context of several state-of-the-art approaches for both linear and nonlinear system ID. In the former, we analyze least squares approaches for Markov parameter estimation, and in the latter, we analyze the multiple shooting approach. We demonstrate the limitations of the optimization problems posed by these existing methods by showing that they can be seen as special cases of the proposed optimization objective under certain simplifying assumptions: conditional independence of data and zero model error. Furthermore, we observe that our proposed approach has improved smoothness and inherent regularization that make it well-suited for system ID and provide mathematical explanations for these characteristics' origins. Finally, numerical simulations demonstrate a mean squared error over 8.7 times lower compared to multiple shooting when data are noisy and/or sparse. Moreover, the proposed approach can identify accurate and generalizable models even when there are more parameters than data or when the underlying system exhibits chaotic behavior.

Bayesian networks (BNs) are a probabilistic graphical model widely used for representing expert knowledge and reasoning under uncertainty. Traditionally, they are based on directed acyclic graphs that capture dependencies between random variables. However, directed cycles can naturally arise when cross-dependencies between random variables exist, e.g., for modeling feedback loops. Existing methods to deal with such cross-dependencies usually rely on reductions to BNs without cycles. These approaches are fragile to generalize, since their justifications are intermingled with additional knowledge about the application context. In this paper, we present a foundational study regarding semantics for cyclic BNs that are generic and conservatively extend the cycle-free setting. First, we propose constraint-based semantics that specify requirements for full joint distributions over a BN to be consistent with the local conditional probabilities and independencies. Second, two kinds of limit semantics that formalize infinite unfolding approaches are introduced and shown to be computable by a Markov chain construction.

Understanding and predicting highway lane-change maneuvers is essential for driving modeling and its automation. The development of data-based lane-changing decision-making algorithms is nowadays in full expansion. We compare empirically in this article different machine and ensemble learning classification techniques to the MOBIL rule-based model using trajectory data of European two-lane highways. The analysis relies on instantaneous measurements of up to twenty-four spatial-temporal variables with the four neighboring vehicles on current and adjacent lanes. Preliminary descriptive investigations by principal component and logistic analyses allow identifying main variables intending a driver to change lanes. We predict two types of discretionary lane-change maneuvers: overtaking (from the slow to the fast lane) and fold-down (from the fast to the slow lane). The prediction accuracy is quantified using total, lane-changing and lane-keeping errors and associated receiver operating characteristic curves. The benchmark analysis includes logistic model, linear discriminant, decision tree, na\"ive Bayes classifier, support vector machine, neural network machine learning algorithms, and up to ten bagging and stacking ensemble learning meta-heuristics. If the rule-based model provides limited predicting accuracy, the data-based algorithms, devoid of modeling bias, allow significant prediction improvements. Cross validations show that selected neural networks and stacking algorithms allow predicting from a single observation both fold-down and overtaking maneuvers up to four seconds in advance with high accuracy.

We propose a novel online and adaptive truncation method for differentially private Bayesian online estimation of a static parameter regarding a population. We assume that sensitive information from individuals is collected sequentially and the inferential aim is to estimate, on-the-fly, a static parameter regarding the population to which those individuals belong. We propose sequential Monte Carlo to perform online Bayesian estimation. When individuals provide sensitive information in response to a query, it is necessary to perturb it with privacy-preserving noise to ensure the privacy of those individuals. The amount of perturbation is proportional to the sensitivity of the query, which is determined usually by the range of the queried information. The truncation technique we propose adapts to the previously collected observations to adjust the query range for the next individual. The idea is that, based on previous observations, we can carefully arrange the interval into which the next individual's information is to be truncated before being perturbed with privacy-preserving noise. In this way, we aim to design predictive queries with small sensitivity, hence small privacy-preserving noise, enabling more accurate estimation while maintaining the same level of privacy. To decide on the location and the width of the interval, we use an exploration-exploitation approach a la Thompson sampling with an objective function based on the Fisher information of the generated observation. We show the merits of our methodology with numerical examples.

We propose a novel definition of Shapley values with uncertain value functions based on first principles using probability theory. Such uncertain value functions can arise in the context of explainable machine learning as a result of non-deterministic algorithms. We show that random effects can in fact be absorbed into a Shapley value with a noiseless but shifted value function. Hence, Shapley values with uncertain value functions can be used in analogy to regular Shapley values. However, their reliable evaluation typically requires more computational effort.