Message-passing graph neural networks (MPNNs) have emerged as the leading approach for machine learning on graphs, attracting significant attention in recent years. While a large set of works explored the expressivity of MPNNs, i.e., their ability to separate graphs and approximate functions over them, comparatively less attention has been directed toward investigating their generalization abilities, i.e., making meaningful predictions beyond the training data. Here, we systematically review the existing literature on the generalization abilities of MPNNs. We analyze the strengths and limitations of various studies in these domains, providing insights into their methodologies and findings. Furthermore, we identify potential avenues for future research, aiming to deepen our understanding of the generalization abilities of MPNNs.

Nonlinear Bayesian update for a prior ensemble is proposed to extend traditional ensemble Kalman filtering to settings characterized by non-Gaussian priors and nonlinear measurement operators. In this framework, the observed component is first denoised via a standard Kalman update, while the unobserved component is estimated using a nonlinear regression approach based on kernel density estimation. The method incorporates a subsampling strategy to ensure stability and, when necessary, employs unsupervised clustering to refine the conditional estimate. Numerical experiments on Lorenz systems and a PDE-constrained inverse problem illustrate that the proposed nonlinear update can reduce estimation errors compared to standard linear updates, especially in highly nonlinear scenarios.

Gaussian processes (GPs) are powerful probabilistic models that define flexible priors over functions, offering strong interpretability and uncertainty quantification. However, GP models often rely on simple, stationary kernels which can lead to suboptimal predictions and miscalibrated uncertainty estimates, especially in nonstationary real-world applications. In this paper, we introduce SEEK, a novel class of learnable kernels to model complex, nonstationary functions via GPs. Inspired by artificial neurons, SEEK is derived from first principles to ensure symmetry and positive semi-definiteness, key properties of valid kernels. The proposed method achieves flexible and adaptive nonstationarity by learning a mapping from a set of base kernels. Compared to existing techniques, our approach is more interpretable and much less prone to overfitting. We conduct comprehensive sensitivity analyses and comparative studies to demonstrate that our approach is not robust to only many of its design choices, but also outperforms existing stationary/nonstationary kernels in both mean prediction accuracy and uncertainty quantification.

Street scene datasets, collected from Street View or dashboard cameras, offer a promising means of detecting urban objects and incidents like street flooding. However, a major challenge in using these datasets is their lack of reliable labels: there are myriad types of incidents, many types occur rarely, and ground-truth measures of where incidents occur are lacking. Here, we propose BayFlood, a two-stage approach which circumvents this difficulty. First, we perform zero-shot classification of where incidents occur using a pretrained vision-language model (VLM). Second, we fit a spatial Bayesian model on the VLM classifications. The zero-shot approach avoids the need to annotate large training sets, and the Bayesian model provides frequent desiderata in urban settings - principled measures of uncertainty, smoothing across locations, and incorporation of external data like stormwater accumulation zones. We comprehensively validate this two-stage approach, showing that VLMs provide strong zero-shot signal for floods across multiple cities and time periods, the Bayesian model improves out-of-sample prediction relative to baseline methods, and our inferred flood risk correlates with known external predictors of risk. Having validated our approach, we show it can be used to improve urban flood detection: our analysis reveals 113,738 people who are at high risk of flooding overlooked by current methods, identifies demographic biases in existing methods, and suggests locations for new flood sensors. More broadly, our results showcase how Bayesian modeling of zero-shot LM annotations represents a promising paradigm because it avoids the need to collect large labeled datasets and leverages the power of foundation models while providing the expressiveness and uncertainty quantification of Bayesian models.

Modern neural network architectures have achieved remarkable accuracies but remain highly dependent on their training data, often lacking interpretability in their learned mappings. While effective on large datasets, they tend to overfit on smaller ones. Probabilistic neural networks, such as those utilizing variational inference, address this limitation by incorporating uncertainty estimation through weight distributions rather than point estimates. However, standard variational inference often relies on a single-density approximation, which can lead to poor posterior estimates and hinder model performance. We propose Boosted Bayesian Neural Networks (BBNN), a novel approach that enhances neural network weight distribution approximations using Boosting Variational Inference (BVI). By iteratively constructing a mixture of densities, BVI expands the approximating family, enabling a more expressive posterior that leads to improved generalization and uncertainty estimation. While this approach increases computational complexity, it significantly enhances accuracy an essential tradeoff, particularly in high-stakes applications such as medical diagnostics, where false negatives can have severe consequences. Our experimental results demonstrate that BBNN achieves ~5% higher accuracy compared to conventional neural networks while providing superior uncertainty quantification. This improvement highlights the effectiveness of leveraging a mixture-based variational family to better approximate the posterior distribution, ultimately advancing probabilistic deep learning.

Traditional statistical inference techniques, such as likelihood ratio tests, have seen renewed interest in recent years, driven in part by the growing emphasis on methodologies based on e-values and e-processes, rather than conventional p-values. Unlike p-values, e-values possess several properties that make them particularly appealing for modern data science applications. In particular, e-value-based methods have played an instrumental role in advancing multiple and safe testing (Grünwald et al., 2020; Vovk and Wang, 2021; Shafer, 2021; Wang and Ramdas, 2022), anytime-valid inference (Waudby-Smith and Ramdas, 2024), and asymptotic confidence sequences (Waudby-Smith et al., 2024). This list is far from exhaustive, and we refer to Ramdas et al. (2023) for a broader overview of recent developments. This manuscript revisits the work of Wasserman et al. (2020), who introduced universal inference, a general hypothesis testing framework based on split likelihood ratio statistics, which is also an e-value. This framework provides simple procedures for many complex composite testing problems that previously lacked actionable solutions, such as testing logconcavity (Dunn et al., 2024) and causal inference under unknown causal structures (Strieder et al., 2021), among others. Specifically, universal inference combines the classical idea of sample splitting (Cox, 1975) and Markov's inequality to establish finite-sample validity. The procedure follows three steps.

Methods to quantify uncertainty in predictions from arbitrary models are in demand in high-stakes domains like medicine and finance. Conformal prediction has emerged as a popular method for producing a set of predictions with specified average coverage, in place of a single prediction and confidence value. However, the value of conformal prediction sets to assist human decisions remains elusive due to the murky relationship between coverage guarantees and decision makers' goals and strategies. How should we think about conformal prediction sets as a form of decision support? We outline a decision theoretic framework for evaluating predictive uncertainty as informative signals, then contrast what can be said within this framework about idealized use of calibrated probabilities versus conformal prediction sets. Informed by prior empirical results and theories of human decisions under uncertainty, we formalize a set of possible strategies by which a decision maker might use a prediction set. We identify ways in which conformal prediction sets and posthoc predictive uncertainty quantification more broadly are in tension with common goals and needs in human-AI decision making. We give recommendations for future research in predictive uncertainty quantification to support human decision makers.

--This paper addresses the integration of additional information sources into a Bayesian optimization framework while ensuring that safety constraints are satisfied. The interdependencies between these information sources are modeled using an unknown correlation matrix. We explore how uniform error bounds must be adjusted to maintain constraint satisfaction throughout the optimization process, considering both Bayesian and frequentist statistical perspectives. This is achieved by appropriately scaling the error bounds based on a confidence interval that can be estimated from the data. Furthermore, the efficacy of the proposed approach is demonstrated through experiments on two benchmark functions and a controller parameter optimization problem. Our results highlight a significant improvement in sample efficiency, demonstrating the method's suitability for optimizing expensive-to-evaluate functions. Many practical optimization problems can be formulated as the optimization of a black-box function, e. g., because of their complex underlying physics or the requirement of impractical identification processes. Black-box optimization algorithms bypass the need of models for optimizations. In essence, these algorithms sequentially evaluate the black-box function for some input while reducing the cost. In the last decade, Bayesian optimization (BO) has emerged as a promising method for solving exactly this set of problems. This method involves constructing a probabilistic surrogate model of an arbitrary objective function with minimal assumptions. The utilization of Gaussian processes (GPs) enables the incorporation of prior knowledge about the objective function, making BO particularly well-suited for scenarios where function evaluations are costly and observations may be noisy. As a simple example of BO, consider the optimization of a PID controller for unit step reference tracking, where the plant dynamics are unknown. A potential cost function that measures tracking accuracy could be the mean-squared error of the plant output and the step reference for a designated time window. The black-box function is now the function that maps the PID parameters to the image of the cost function. An evaluation corresponds to running the step response of the system with the specified PID parameters.

Despite their groundbreaking performance, state-of-the-art autonomous agents can misbehave when training and environmental conditions become inconsistent, with minor mismatches leading to undesirable behaviors or even catastrophic failures. Robustness towards these training/environment ambiguities is a core requirement for intelligent agents and its fulfillment is a long-standing challenge when deploying agents in the real world. Here, departing from mainstream views seeking robustness through training, we introduce DR-FREE, a free energy model that installs this core property by design. It directly wires robustness into the agent decision-making mechanisms via free energy minimization. By combining a robust extension of the free energy principle with a novel resolution engine, DR-FREE returns a policy that is optimal-yet-robust against ambiguity. Moreover, for the first time, it reveals the mechanistic role of ambiguity on optimal decisions and requisite Bayesian belief updating. We evaluate DR-FREE on an experimental testbed involving real rovers navigating an ambiguous environment filled with obstacles. Across all the experiments, DR-FREE enables robots to successfully navigate towards their goal even when, in contrast, standard free energy minimizing agents that do not use DR-FREE fail. In short, DR-FREE can tackle scenarios that elude previous methods: this milestone may inspire both deployment in multi-agent settings and, at a perhaps deeper level, the quest for a biologically plausible explanation of how natural agents - with little or no training - survive in capricious environments.

Ensuring transparency in educational assessment is increasingly critical, particularly post-pandemic, as demand grows for fairer and more reliable evaluation methods. Comparative Judgement (CJ) offers a promising alternative to traditional assessments, yet concerns remain about its perceived opacity. This paper examines how Bayesian Comparative Judgement (BCJ) enhances transparency by integrating prior information into the judgement process, providing a structured, data-driven approach that improves interpretability and accountability. BCJ assigns probabilities to judgement outcomes, offering quantifiable measures of uncertainty and deeper insights into decision confidence. By systematically tracking how prior data and successive judgements inform final rankings, BCJ clarifies the assessment process and helps identify assessor disagreements. Multi-criteria BCJ extends this by evaluating multiple learning outcomes (LOs) independently, preserving the richness of CJ while producing transparent, granular rankings aligned with specific assessment goals. It also enables a holistic ranking derived from individual LOs, ensuring comprehensive evaluations without compromising detailed feedback. Using a real higher education dataset with professional markers in the UK, we demonstrate BCJ's quantitative rigour and ability to clarify ranking rationales. Through qualitative analysis and discussions with experienced CJ practitioners, we explore its effectiveness in contexts where transparency is crucial, such as high-stakes national assessments. We highlight the benefits and limitations of BCJ, offering insights into its real-world application across various educational settings.