The development of safety validation methods is essential for the safe deployment and operation of Automated Driving Systems (ADSs). One of the goals of safety validation is to prospectively evaluate the risk of an ADS dealing with real-world traffic. Scenario-based assessment is a widely-used approach, where test cases are derived from real-world driving data. To allow for a quantitative analysis of the system performance, the exposure of the scenarios must be accurately estimated. The exposure of scenarios at parameter level is expressed using a Probability Density Function (PDF). However, assumptions about the PDF, such as parameter independence, can introduce errors, while avoiding assumptions often leads to oversimplified models with limited parameters to mitigate the curse of dimensionality. This paper considers the use of Normalizing Flows (NF) for estimating the PDF of the parameters. NF are a class of generative models that transform a simple base distribution into a complex one using a sequence of invertible and differentiable mappings, enabling flexible, high-dimensional density estimation without restrictive assumptions on the PDF's shape. We demonstrate the effectiveness of NF in quantifying risk and risk uncertainty of an ADS, comparing its performance with Kernel Density Estimation (KDE), a traditional method for non-parametric PDF estimation. While NF require more computational resources compared to KDE, NF is less sensitive to the curse of dimensionality. As a result, NF can improve risk uncertainty estimation, offering a more precise assessment of an ADS's safety. This work illustrates the potential of NF in scenario-based safety. Future work involves experimenting more with using NF for scenario generation and optimizing the NF architecture, transformation types, and training hyperparameters to further enhance their applicability.

The scalable Markov chain Monte Carlo (MCMC) algorithms that underpin modern Bayesian machine learning, such as Stochastic Gradient Langevin Dynamics (SGLD), sacrifice asymptotic exactness for computational speed, creating a critical diagnostic gap: traditional sample quality measures fail catastrophically when applied to biased samplers. While powerful Stein-based diagnostics can detect distributional mismatches, they provide no direct assessment of dependence structure, often the primary inferential target in multivariate problems. We introduce the Copula Discrepancy (CD), a principled and computationally efficient diagnostic that leverages Sklar's theorem to isolate and quantify the fidelity of a sample's dependence structure independent of its marginals. Our theoretical framework provides the first structure-aware diagnostic specifically designed for the era of approximate inference. Empirically, we demonstrate that a moment-based CD dramatically outperforms standard diagnostics like effective sample size for hyperparameter selection in biased MCMC, correctly identifying optimal configurations where traditional methods fail. Furthermore, our robust MLE-based variant can detect subtle but critical mismatches in tail dependence that remain invisible to rank correlation-based approaches, distinguishing between samples with identical Kendall's tau but fundamentally different extreme-event behavior. With computational overhead orders of magnitude lower than existing Stein discrepancies, the CD provides both immediate practical value for MCMC practitioners and a theoretical foundation for the next generation of structure-aware sample quality assessment.

InsurTech refers to the use of state-of-the-art technology, including both emerging hardware and software, to address inefficiencies across the insurance value chain and further explore new opportunities to reshape traditional business operations. InsurTech encompasses a broad spectrum of technology-driven innovations, including, but not limited to, telematics, usage-based insurance, and the integration of Internet of Things (IoT) sensors. In this study, we focus on a specific class of InsurTech, an Insurtech data vendor, that provides insurance companies with next-generation data solutions. We leverage new and diverse external data sources, such as social media data and online content, to enrich the internal database, thereby empowering actuarial analytics and gaining more accurate insights into risk profiles and policyholder behavior. Specifically, by integrating alternative data sources beyond traditional information, insurance companies can uncover previously unrecognized risk factors, reduce bias in existing features, and identify more accurate risk exposures based on the operational characteristics of the insured entities.

We develop a unified Data Processing Inequality PAC-Bayesian framework -- abbreviated DPI-PAC-Bayesian -- for deriving generalization error bounds in the supervised learning setting. By embedding the Data Processing Inequality (DPI) into the change-of-measure technique, we obtain explicit bounds on the binary Kullback-Leibler generalization gap for both Rényi divergence and any $f$-divergence measured between a data-independent prior distribution and an algorithm-dependent posterior distribution. We present three bounds derived under our framework using Rényi, Hellinger \(p\) and Chi-Squared divergences. Additionally, our framework also demonstrates a close connection with other well-known bounds. When the prior distribution is chosen to be uniform, our bounds recover the classical Occam's Razor bound and, crucially, eliminate the extraneous \(\log(2\sqrt{n})/n\) slack present in the PAC-Bayes bound, thereby achieving tighter results. The framework thus bridges data-processing and PAC-Bayesian perspectives, providing a flexible, information-theoretic tool to construct generalization guarantees.

Graph neural networks (GNNs) excel at predictive tasks on graph-structured data but often lack the ability to incorporate symbolic domain knowledge and perform general reasoning. Relational Bayesian Networks (RBNs), in contrast, enable fully generative probabilistic modeling over graph-like structures and support rich symbolic knowledge and probabilistic inference. This paper presents a neuro-symbolic framework that seamlessly integrates GNNs into RBNs, combining the learning strength of GNNs with the flexible reasoning capabilities of RBNs. We develop two implementations of this integration: one compiles GNNs directly into the native RBN language, while the other maintains the GNN as an external component. Both approaches preserve the semantics and computational properties of GNNs while fully aligning with the RBN modeling paradigm. We also propose a maximum a-posteriori (MAP) inference method for these neuro-symbolic models. To demonstrate the framework's versatility, we apply it to two distinct problems. First, we transform a GNN for node classification into a collective classification model that explicitly models homo- and heterophilic label patterns, substantially improving accuracy. Second, we introduce a multi-objective network optimization problem in environmental planning, where MAP inference supports complex decision-making. Both applications include new publicly available benchmark datasets. This work introduces a powerful and coherent neuro-symbolic approach to graph data, bridging learning and reasoning in ways that enable novel applications and improved performance across diverse tasks.

In the field of Machine Learning (ML) and data-driven applications, one of the significant challenge is the change in data distribution between the training and deployment stages, commonly known as distribution shift. This paper outlines different mechanisms for handling two main types of distribution shifts: (i) Covariate shift: where the value of features or covariates change between train and test data, and (ii) Concept/Semantic-shift: where model experiences shift in the concept learned during training due to emergence of novel classes in the test phase. We sum up our contributions in three folds. First, we formalize distribution shifts, recite on how the conventional method fails to handle them adequately and urge for a model that can simultaneously perform better in all types of distribution shifts. Second, we discuss why handling distribution shifts is important and provide an extensive review of the methods and techniques that have been developed to detect, measure, and mitigate the effects of these shifts. Third, we discuss the current state of distribution shift handling mechanisms and propose future research directions in this area. Overall, we provide a retrospective synopsis of the literature in the distribution shift, focusing on OOD data that had been overlooked in the existing surveys.

Embodied intelligence posits that cognitive capabilities fundamentally emerge from - and are shaped by - an agent's real-time sensorimotor interactions with its environment. Such adaptive behavior inherently requires continuous inference under uncertainty. Bayesian statistics offers a principled probabilistic framework to address this challenge by representing knowledge as probability distributions and updating beliefs in response to new evidence. The core computational processes underlying embodied intelligence - including perception, action selection, learning, and even higher-level cognition - can be effectively understood and modeled as forms of Bayesian inference. Despite the deep conceptual connection between Bayesian statistics and embodied intelligence, Bayesian principles have not been widely or explicitly applied in today's embodied intelligence systems. In this work, we examine both Bayesian and contemporary embodied intelligence approaches through two fundamental lenses: search and learning - the two central themes in modern AI, as highlighted in Rich Sutton's influential essay "The Bitter Lesson". This analysis sheds light on why Bayesian inference has not played a central role in the development of modern embodied intelligence. At the same time, it reveals that current embodied intelligence systems remain largely confined to closed-physical-world environments, and highlights the potential for Bayesian methods to play a key role in extending these systems toward truly open physical-world embodied intelligence.

This paper studies fine-grained singular subspace estimation in the matrix denoising model where a deterministic low-rank signal matrix is additively perturbed by a stochastic matrix of Gaussian noise. We establish that the maximum Euclidean row norm (i.e., the two-to-infinity norm) of the aligned difference between the leading sample and population singular vectors approaches the Gumbel distribution in the large-matrix limit, under suitable signal-to-noise conditions and after appropriate centering and scaling. We apply our novel asymptotic distributional theory to test hypotheses of low-rank signal structure encoded in the leading singular vectors and their corresponding principal subspace. We provide de-biased estimators for the corresponding nuisance signal singular values and show that our proposed plug-in test statistic has desirable properties. Notably, compared to using the Frobenius norm subspace distance, our test statistic based on the two-to-infinity norm has higher power to detect structured alternatives that differ from the null in only a few matrix entries or rows. Our main results are obtained by a novel synthesis of and technical analysis involving entrywise matrix perturbation analysis, extreme value theory, saddle point approximation methods, and random matrix theory. Our contributions complement the existing literature for matrix denoising focused on minimaxity, mean squared error analysis, unitarily invariant distances between subspaces, component-wise asymptotic distributional theory, and row-wise uniform error bounds. Numerical simulations illustrate our main results and demonstrate the robustness properties of our testing procedure to non-Gaussian noise distributions.

While achieving exact conditional coverage in conformal prediction is unattainable without making strong, untestable regularity assumptions, the promise of conformal prediction hinges on finding approximations to conditional guarantees that are realizable in practice. A promising direction for obtaining conditional dependence for conformal sets--in particular capturing heteroskedasticity--is through estimating the conditional density $\mathbb{P}_{Y|X}$ and conformalizing its level sets. Previous work in this vein has focused on nonconformity scores based on the empirical cumulative distribution function (CDF). Such scores are, however, computationally costly, typically requiring expensive sampling methods. To avoid the need for sampling, we observe that the CDF-based score reduces to a Mahalanobis distance in the case of Gaussian scores, yielding a closed-form expression that can be directly conformalized. Moreover, the use of a Gaussian-based score opens the door to a number of extensions of the basic conformal method; in particular, we show how to construct conformal sets with missing output values, refine conformal sets as partial information about $Y$ becomes available, and construct conformal sets on transformations of the output space. Finally, empirical results indicate that our approach produces conformal sets that more closely approximate conditional coverage in multivariate settings compared to alternative methods.

Uncertainty quantification is an important prerequisite for the deployment of deep learning models in safety-critical areas. Yet, this hinges on the uncertainty estimates being useful to the extent the prediction intervals are well-calibrated and sharp. In the absence of inherent uncertainty estimates (e.g. pretrained models predicting only point estimates), popular approaches that operate post-hoc include Laplace's method and split conformal prediction (split-CP). However, Laplace's method can be miscalibrated when the model is misspecified and split-CP requires sample splitting, and thus comes at the expense of statistical efficiency. In this work, we construct prediction intervals for neural network regressors post-hoc without held-out data. This is achieved by approximating the full conformal prediction method (full-CP). Whilst full-CP nominally requires retraining the model for every test point and candidate label, we propose to train just once and locally perturb model parameters using Gauss-Newton influence to approximate the effect of retraining. Coupled with linearization of the network, we express the absolute residual nonconformity score as a piecewise linear function of the candidate label allowing for an efficient procedure that avoids the exhaustive search over the output space. On standard regression benchmarks and bounding box localization, we show the resulting prediction intervals are locally-adaptive and often tighter than those of split-CP.