Out-of-Distribution (OOD) detection is a cornerstone for the safe deployment of AI systems in the open world. However, existing methods treat OOD detection as a binary classification problem, a cognitive flattening that fails to distinguish between semantically close (Near-OOD) and distant (Far-OOD) unknown risks. This limitation poses a significant safety bottleneck in applications requiring fine-grained risk stratification. To address this, we propose a paradigm shift from a conventional probabilistic view to a principled information-theoretic framework. We formalize the core task as quantifying the Semantic Surprise of a new sample and introduce a novel ternary classification challenge: In-Distribution (ID) vs. Near-OOD vs. Far-OOD. The theoretical foundation of our work is the concept of Low-Entropy Semantic Manifolds, which are explicitly structured to reflect the data's intrinsic semantic hierarchy. To construct these manifolds, we design a Hierarchical Prototypical Network. We then introduce the Semantic Surprise Vector (SSV), a universal probe that decomposes a sample's total surprise into three complementary and interpretable dimensions: conformity, novelty, and ambiguity. To evaluate performance on this new task, we propose the Normalized Semantic Risk (nSR), a cost-sensitive metric. Experiments demonstrate that our framework not only establishes a new state-of-the-art (sota) on the challenging ternary task, but its robust representations also achieve top results on conventional binary benchmarks, reducing the False Positive Rate by over 60% on datasets like LSUN.

Uncertainty quantification is critical for ensuring robustness in high-stakes machine learning applications. We introduce HybridFlow, a modular hybrid architecture that unifies the modeling of aleatoric and epistemic uncertainty by combining a Conditional Masked Autoregressive normalizing flow for estimating aleatoric uncertainty with a flexible probabilistic predictor for epistemic uncertainty. The framework supports integration with any probabilistic model class, allowing users to easily adapt HybridFlow to existing architectures without sacrificing predictive performance. HybridFlow improves upon previous uncertainty quantification frameworks across a range of regression tasks, such as depth estimation, a collection of regression benchmarks, and a scientific case study of ice sheet emulation. We also provide empirical results of the quantified uncertainty, showing that the uncertainty quantified by HybridFlow is calibrated and better aligns with model error than existing methods for quantifying aleatoric and epistemic uncertainty. HybridFlow addresses a key challenge in Bayesian deep learning, unifying aleatoric and epistemic uncertainty modeling in a single robust framework.

Asymptotic efficiency theory is one of the pillars in the foundations of modern mathematical statistics. Not only does it serve as a rigorous theoretical benchmark for evaluating statistical methods, but it also sheds light on how to develop and unify novel statistical procedures. For example, the calculus of influence functions has led to many important statistical breakthroughs in the past decades. Responding to the pressing challenge of analyzing increasingly complex datasets, particularly those with non-Euclidean/nonlinear structures, many novel statistical models and methods have been proposed in recent years. However, the existing efficiency theory is not always readily applicable to these cases, as the theory was developed, for the most part, under the often neglected premise that both the sample space and the parameter space are normed linear spaces. As a consequence, efficiency results outside normed linear spaces are quite rare and isolated, obtained on a case-by-case basis. This paper aims to develop a more unified asymptotic efficiency theory, allowing the sample space, the parameter space, or both to be Riemannian manifolds satisfying certain regularity conditions. We build a vocabulary that helps translate essential concepts in efficiency theory from normed linear spaces to Riemannian manifolds, such as (locally) regular estimators, differentiable functionals, etc. Efficiency bounds are established under conditions parallel to those for normed linear spaces. We also demonstrate the conceptual advantage of the new framework by applying it to two concrete examples in statistics: the population Frechet mean and the regression coefficient vector of Single-Index Models.

We study neural network compressibility by using singular learning theory to extend the minimum description length (MDL) principle to singular models like neural networks. Through extensive experiments on the Pythia suite with quantization, factorization, and other compression techniques, we find that complexity estimates based on the local learning coefficient (LLC) are closely, and in some cases, linearly correlated with compressibility. Our results provide a path toward rigorously evaluating the limits of model compression.

We develop a unified statistical framework for softmax-gated Gaussian mixture of experts (SGMoE) that addresses three long-standing obstacles in parameter estimation and model selection: (i) non-identifiability of gating parameters up to common translations, (ii) intrinsic gate-expert interactions that induce coupled differential relations in the likelihood, and (iii) the tight numerator-denominator coupling in the softmax-induced conditional density. Our approach introduces Voronoi-type loss functions aligned with the gate-partition geometry and establishes finite-sample convergence rates for the maximum likelihood estimator (MLE). In over-specified models, we reveal a link between the MLE's convergence rate and the solvability of an associated system of polynomial equations characterizing near-nonidentifiable directions. For model selection, we adapt dendrograms of mixing measures to SGMoE, yielding a consistent, sweep-free selector of the number of experts that attains pointwise-optimal parameter rates under overfitting while avoiding multi-size training. Simulations on synthetic data corroborate the theory, accurately recovering the expert count and achieving the predicted rates for parameter estimation while closely approximating the regression function. Under model misspecification (e.g., $ε$-contamination), the dendrogram selection criterion is robust, recovering the true number of mixture components, while the Akaike information criterion, the Bayesian information criterion, and the integrated completed likelihood tend to overselect as sample size grows. On a maize proteomics dataset of drought-responsive traits, our dendrogram-guided SGMoE selects two experts, exposes a clear mixing-measure hierarchy, stabilizes the likelihood early, and yields interpretable genotype-phenotype maps, outperforming standard criteria without multi-size training.

With advancements in reasoning capabilities, Large Language Models (LLMs) are increasingly employed for automated judgment tasks. While LLMs-as-Judges offer promise in automating evaluations, current approaches often rely on simplistic aggregation methods (e.g., majority voting), which can fail even when individual agents provide correct answers. To address this, we propose a multi-agent debate judge framework where agents collaboratively reason and iteratively refine their responses. We formalize the debate process mathematically, analyzing agent interactions and proving that debate amplifies correctness compared to static ensembles. To enhance efficiency, we introduce a stability detection mechanism that models judge consensus dynamics via a time-varying Beta-Binomial mixture, with adaptive stopping based on distributional similarity (Kolmogorov-Smirnov test). This mechanism models the judges' collective correct rate dynamics using a time-varying mixture of Beta-Binomial distributions and employs an adaptive stopping criterion based on distributional similarity (Kolmogorov-Smirnov statistic). Experiments across multiple benchmarks and models demonstrate that our framework improves judgment accuracy over majority voting while maintaining computational efficiency.

Accurate intraday forecasts are essential for power system operations, complementing day-ahead forecasts that gradually lose relevance as new information becomes available. This paper introduces a Bayesian updating mechanism that converts fully probabilistic day-ahead forecasts into intraday forecasts without retraining or re-inference. The approach conditions the Gaussian mixture output of a conditional variational autoencoder-based forecaster on observed measurements, yielding an updated distribution for the remaining horizon that preserves its probabilistic structure. This enables consistent point, quantile, and ensemble forecasts while remaining computationally efficient and suitable for real-time applications. Experiments on household electricity consumption and photovoltaic generation datasets demonstrate that the proposed method improves forecast accuracy up to 25% across likelihood-, sample-, quantile-, and point-based metrics. The largest gains occur in time steps with strong temporal correlation to observed data, and the use of pattern dictionary-based covariance structures further enhances performance. The results highlight a theoretically grounded framework for intraday forecasting in modern power systems.

Standard supervised learners attempt to learn a model from a labeled dataset. Given a small set of labeled instances, and a pool of unlabeled instances, a budgeted learner can use its given budget to pay to acquire the labels of some unlabeled instances, which it can then use to produce a model. Here, we explore budgeted learning in the context of survival datasets, which include (right) censored instances, where we know only a lower bound on an instance's time-to-event. Here, that learner can pay to (partially) label a censored instance -- e.g., to acquire the actual time for an instance [perhaps go from (3 yr, censored) to (7.2 yr, uncensored)], or other variants [e.g., learn about one more year, so go from (3 yr, censored) to either (4 yr, censored) or perhaps (3.2 yr, uncensored)]. This serves as a model of real world data collection, where follow-up with censored patients does not always lead to uncensoring, and how much information is given to the learner model during data collection is a function of the budget and the nature of the data itself. We provide both experimental and theoretical results for how to apply state-of-the-art budgeted learning algorithms to survival data and the respective limitations that exist in doing so. Our approach provides bounds and time complexity asymptotically equivalent to the standard active learning method BatchBALD. Moreover, empirical analysis on several survival tasks show that our model performs better than other potential approaches on several benchmarks.

Generative classifiers, which leverage conditional generative models for classification, have recently demonstrated desirable properties such as robustness to distribution shifts. However, recent progress in this area has been largely driven by diffusion-based models, whose substantial computational cost severely limits scalability. This exclusive focus on diffusion-based methods has also constrained our understanding of generative classifiers. In this work, we propose a novel generative classifier built on recent advances in visual autoregressive (V AR) modeling, which offers a new perspective for studying generative classifiers. Moreover, we show that the V ARbased method exhibits fundamentally different properties from diffusion-based methods. In particular, due to its tractable likelihood, the V AR-based classifier enables visual explainability via token-wise mutual information and demonstrates inherent resistance to catastrophic forgetting in class-incremental learning tasks. Generative models are trained to directly capture the underlying data distribution of a given dataset, which enables a wide range of applications such as image generation (Han et al., 2025), image editing (Mu et al., 2025), and data augmentation (Trabucco et al., 2023). Given this expressive capability, a natural question arises: Can we leverage these powerful generative models for classification? This question has motivated a line of research on the "Generative Classifier."

The rapid advancement of large language models (LLMs) has transformed the landscape of natural language processing, enabling breakthroughs across a wide range of areas including question answering, machine translation, and text summarization. Yet, their deployment in real-world applications has raised concerns over reliability and trustworthiness, as LLMs remain prone to hallucinations that produce plausible but factually incorrect outputs. Uncertainty quantification (UQ) has emerged as a central research direction to address this issue, offering principled measures for assessing the trustworthiness of model generations. We begin by introducing the foundations of UQ, from its formal definition to the traditional distinction between epistemic and aleatoric uncertainty, and then highlight how these concepts have been adapted to the context of LLMs. Building on this, we examine the role of UQ in hallucination detection, where quantifying uncertainty provides a mechanism for identifying unreliable generations and improving reliability. We systematically categorize a wide spectrum of existing methods along multiple dimensions and present empirical results for several representative approaches. Finally, we discuss current limitations and outline promising future research directions, providing a clearer picture of the current landscape of LLM UQ for hallucination detection.