These notes provide a self-contained introduction to kernel methods and their geometric foundations in machine learning. Starting from the construction of Hilbert spaces, we develop the theory of positive definite kernels, reproducing kernel Hilbert spaces (RKHS), and Hilbert-Schmidt operators, emphasizing their role in statistical estimation and representation of probability measures. Classical concepts such as covariance, regression, and information measures are revisited through the lens of Hilbert space geometry. We also introduce kernel density estimation, kernel embeddings of distributions, and the Maximum Mean Discrepancy (MMD). The exposition is designed to serve as a foundation for more advanced topics, including Gaussian processes, kernel Bayesian inference, and functional analytic approaches to modern machine learning.

Bayesian networks (BNs) are a widely used class of probabilistic graphical models employed in numerous application domains. However, inferring the network's graphical structure from data remains challenging. Bayesian structure learners approach this problem by inferring a posterior distribution over the possible directed acyclic graphs underlying the BN. The inference process often requires marginalizing over probability distributions, which is typically done using dynamic programming methods that restrict the set of possible parents for each node. Instead, we present a novel method that utilizes tractable probabilistic circuits to circumvent this restriction. This method utilizes a new learning routine that trains these circuits on both the original distribution and marginal queries. The architecture of probabilistic circuits then inherently allows for fast and exact marginalization on the learned distribution. We then show empirically that utilizing our method to answer marginals allows Bayesian structure learners to improve their performance compared to current methods.

Diffusion models (DMs) have emerged as powerful image priors in Bayesian computational imaging. Two primary strategies have been proposed for leveraging DMs in this context: Plug-and-Play methods, which are zero-shot and highly flexible but rely on approximations; and specialized conditional DMs, which achieve higher accuracy and faster inference for specific tasks through supervised training. In this work, we introduce a novel framework that integrates deep unfolding and model distillation to transform a DM image prior into a few-step conditional model for posterior sampling. A central innovation of our approach is the unfolding of a Markov chain Monte Carlo (MCMC) algorithm - specifically, the recently proposed LATINO Langevin sampler (Spagnoletti et al., 2025) - representing the first known instance of deep unfolding applied to a Monte Carlo sampling scheme. We demonstrate our proposed unfolded and distilled samplers through extensive experiments and comparisons with the state of the art, where they achieve excellent accuracy and computational efficiency, while retaining the flexibility to adapt to variations in the forward model at inference time.

Coral bleaching is a major concern for marine ecosystems; more than half of the world's coral reefs have either bleached or died over the past three decades. Increasing sea surface temperatures, along with various spatiotemporal environmental factors, are considered the primary reasons behind coral bleaching. The statistical and machine learning communities have focused on multiple aspects of the environment in detail. However, the literature on various stochastic modeling approaches for assessing coral bleaching is extremely scarce. Data-driven strategies are crucial for effective reef management, and this review article provides an overview of existing statistical and machine learning methods for assessing coral bleaching. Statistical frameworks, including simple regression models, generalized linear models, generalized additive models, Bayesian regression models, spatiotemporal models, and resilience indicators, such as Fisher's Information and Variance Index, are commonly used to explore how different environmental stressors influence coral bleaching. On the other hand, machine learning methods, including random forests, decision trees, support vector machines, and spatial operators, are more popular for detecting nonlinear relationships, analyzing high-dimensional data, and allowing integration of heterogeneous data from diverse sources. In addition to summarizing these models, we also discuss potential data-driven future research directions, with a focus on constructing statistical and machine learning models in specific contexts related to coral bleaching.

X-ray scattering measurements of in situ human brain tissue encode structural signatures of pathological cross-$β$ inclusions, yet systematic exploitation of these data for automated detection remains challenging due to substrate contamination, strong inter-feature correlations, and limited sample sizes. This work develops a three-stage classification framework for identifying cross-$β$ structural inclusions-a hallmark of Alzheimer's disease-in X-ray scattering profiles of post-mortem human brain. Stage 1 employs a Bayes-optimal classifier to separate mica substrate from tissue regions on the basis of their distinct scattering signatures. Stage 2 introduces a multicollinearityaware, class-conditional correlation pruning scheme with formal guarantees on the induced Bayes risk and approximation error, thereby reducing redundancy while retaining class-discriminative information. Stage 3 trains a compact neural network on the pruned feature set to detect the presence or absence of cross-$β$ fibrillar ordering. The top-performing model, optimized with a composite loss combining Focal and Dice objectives, attains a test F1-score of 84.30% using 11 of 211 candidate features and 174 trainable parameters. The overall framework yields an interpretable, theory-grounded strategy for data-limited classification problems involving correlated, high-dimensional experimental measurements, exemplified here by X-ray scattering profiles of neurodegenerative tissue.

The recent success of large language models (LLMs) has sparked a growing interest in training large-scale models. As the model size continues to scale, concerns are growing about the depletion of high-quality, well-curated training data. This has led practitioners to explore training approaches like Federated Learning (FL), which can leverage the abundant data on edge devices while maintaining privacy. However, the decentralization of training datasets in FL introduces challenges to scaling large models, a topic that remains under-explored. This paper fills this gap and provides qualitative insights on generalizing the previous model scaling experience to federated learning scenarios. Specifically, we derive a P AC-Bayes (Probably Approximately Correct Bayesian) upper bound for the generalization error of models trained with stochastic algorithms in federated settings and quantify the impact of distributed training data on the optimal model size by finding the analytic solution of model size that minimizes this bound. Our theoretical results demonstrate that the optimal model size has a negative power law relationship with the number of clients if the total training compute is unchanged. Besides, we also find that switching to FL with the same training compute will inevitably reduce the upper bound of generalization performance that the model can achieve through training, and that estimating the optimal model size in federated scenarios should depend on the average training compute across clients. Furthermore, we also empirically validate the correctness of our results with extensive training runs on different models, network settings, and datasets.

There is growing interest in using machine learning (ML) to support clinical diagnosis, but most approaches rely on static, fully observed datasets and fail to reflect the sequential, resource-aware reasoning clinicians use in practice. Diagnosis remains complex and error prone, especially in high-pressure or resource-limited settings, underscoring the need for frameworks that help clinicians make timely and cost-effective decisions. We propose ACTMED (Adaptive Clinical Test selection via Model-based Experimental Design), a diagnostic framework that integrates Bayesian Experimental Design (BED) with large language models (LLMs) to better emulate real-world diagnostic reasoning. At each step, ACTMED selects the test expected to yield the greatest reduction in diagnostic uncertainty for a given patient. LLMs act as flexible simulators, generating plausible patient state distributions and supporting belief updates without requiring structured, task-specific training data. Clinicians can remain in the loop; reviewing test suggestions, interpreting intermediate outputs, and applying clinical judgment throughout. We evaluate ACTMED on real-world datasets and show it can optimize test selection to improve diagnostic accuracy, interpretability, and resource use. This represents a step toward transparent, adaptive, and clinician-aligned diagnostic systems that generalize across settings with reduced reliance on domain-specific data.

Effective Uncertainty Quantification (UQ) represents a key aspect for reliable deployment of Large Language Models (LLMs) in automated decision-making and beyond. Yet, for LLM generation with multiple choice structure, the state-of-the-art in UQ is still dominated by the naive baseline given by the maximum softmax score. To address this shortcoming, we demonstrate that taking a principled approach via Bayesian statistics leads to improved performance despite leveraging the simplest possible model, namely linear regression. More precisely, we propose to train multiple Bayesian linear models, each predicting the output of a layer given the output of the previous one. Based on the obtained layer-level posterior distributions, we infer the global uncertainty level of the LLM by identifying a sparse combination of distributional features, leading to an efficient UQ scheme. Numerical experiments on various LLMs show consistent improvement over state-of-the-art baselines.

We present Distributional RewArds for Generative OptimizatioN (DRAGON), a versatile framework for fine-tuning media generation models towards a desired outcome. Compared with traditional reinforcement learning with human feedback (RLHF) or pairwise preference approaches such as direct preference optimization (DPO), DRAGON is more flexible. It can optimize reward functions that evaluate either individual examples or distributions of them, making it compatible with a broad spectrum of instance-wise, instance-to-distribution, and distribution-to-distribution rewards. Leveraging this versatility, we construct novel reward functions by selecting an encoder and a set of reference examples to create an exemplar distribution. When cross-modal encoders such as CLAP are used, the reference may be of a different modality (text versus audio). Then, DRAGON gathers online and on-policy generations, scores them with the reward function to construct a positive demonstration set and a negative set, and leverages the contrast between the two finite sets to approximate distributional reward optimization. For evaluation, we fine-tune an audio-domain text-to-music diffusion model with 20 reward functions, including a custom music aesthetics model, CLAP score, Vendi diversity, and Frechet audio distance (FAD). We further compare instance-wise (per-song) and full-dataset FAD settings while ablating multiple FAD encoders and reference sets. Over all 20 target rewards, DRAGON achieves an 81.45% average win rate. Moreover, reward functions based on exemplar sets enhance generations and are comparable to model-based rewards. With an appropriate exemplar set, DRAGON achieves a 60.95% human-voted music quality win rate without training on human preference annotations. DRAGON is a new approach to designing and optimizing reward functions for improving human-perceived quality. Demos at https://ml-dragon.github.io/web

Technological systems increasingly mediate human information exchange, spanning interactions among humans as well as between humans and artificial agents. The unprecedented scale and reliance on information disseminated through these systems substantially expand the scope of information-based influence that can both enable and undermine sound decision-making. Consequently, understanding and protecting decision-making today faces growing challenges, as individuals and organizations must navigate evolving opportunities and information-based threats across varied domains and information environments. While these risks are widely recognized, research remains fragmented: work evaluating information-based threat phenomena has progressed largely in isolation from foundational studies of human information processing. In this review, we synthesize insights from both domains to identify shared cognitive mechanisms that mediate vulnerability to information-based threats and shape behavioral outcomes. Finally, we outline directions for future research aimed at integrating these perspectives, emphasizing the importance of such integration for mitigating human vulnerabilities and aligning human-machine representations.