Electricity price forecasting has become a critical tool for decision-making in energy markets, particularly as the increasing penetration of renewable energy introduces greater volatility and uncertainty. Historically, research in this field has been dominated by point forecasting methods, which provide single-value predictions but fail to quantify uncertainty. However, as power markets evolve due to renewable integration, smart grids, and regulatory changes, the need for probabilistic forecasting has become more pronounced, offering a more comprehensive approach to risk assessment and market participation. This paper presents a review of probabilistic forecasting methods, tracing their evolution from Bayesian and distribution based approaches, through quantile regression techniques, to recent developments in conformal prediction. Particular emphasis is placed on advancements in probabilistic forecasting, including validity-focused methods which address key limitations in uncertainty estimation. Additionally, this review extends beyond the Day-Ahead Market to include the Intra-Day and Balancing Markets, where forecasting challenges are intensified by higher temporal granularity and real-time operational constraints. We examine state of the art methodologies, key evaluation metrics, and ongoing challenges, such as forecast validity, model selection, and the absence of standardised benchmarks, providing researchers and practitioners with a comprehensive and timely resource for navigating the complexities of modern electricity markets.

ABSTRACT Data-driven astrophysics currently relies on the detection and characterisation of correlations between objects' properties, which are then used to test physical theories that make predictions for them. This process fails to utilise information in the data that forms a crucial part of the theories' predictions, namely which variables are directly correlated (as opposed to accidentally correlated through others), the directions of these determinations, and the presence or absence of confounders that correlate variables in the dataset but are themselves absent from it. We propose to recover this information through causal discovery, a well-developed methodology for inferring the causal structure of datasets that is however almost entirely unknown to astrophysics. INTRODUCTION Understanding the physical processes that shape galaxies is a central goal of astrophysics. Empirical progress has traditionally relied on identifying correlations between observed properties, which can then be interpreted in light of theoretical models for galaxy formation and used to constrain them. The advent of large surveys and powerful machine learning techniques has greatly expanded our ability to find such statistical associations, uncovering intricate patterns across high-dimensional parameter spaces. However, correlation alone cannot determine causal influences among variables: which properties are actually responsible for determining others, in what direction this influence goes, and whether there exist confounding variables that are not included in the dataset but influence those that are.

Label learning is a fundamental task in machine learning that aims to construct intelligent models using labeled data, encompassing traditional single-label and multi-label classification models. Traditional methods typically rely on logical labels, such as binary indicators (e.g., "yes/no") that specify whether an instance belongs to a given category. However, in practical applications, label annotations often involve significant uncertainty due to factors such as data noise, inherent ambiguity in the observed entities, and the subjectivity of human annotators. Therefore, representing labels using simplistic binary logic can obscure valuable information and limit the expressiveness of label learning models. To overcome this limitation, this paper introduces the concept of fuzzy labels, grounded in fuzzy set theory, to better capture and represent label uncertainty. We further propose an efficient fuzzy labeling method that mines and generates fuzzy labels from the original data, thereby enriching the label space with more informative and nuanced representations. Based on this foundation, we present fuzzy-label-enhanced algorithms for both single-label and multi-label learning, using the classical K-Nearest Neighbors (KNN) and multi-label KNN algorithms as illustrative examples. Experimental results indicate that fuzzy labels can more effectively characterize the real-world labeling information and significantly enhance the performance of label learning models.

Vector data trading is essential for cross-domain learning with vector databases, yet it remains largely unexplored. We study this problem under online learning, where sellers face uncertain retrieval costs and buyers provide stochastic feedback to posted prices. Three main challenges arise: (1) heterogeneous and partial feedback in configuration learning, (2) variable and complex feedback in pricing learning, and (3) inherent coupling between configuration and pricing decisions. We propose a hierarchical bandit framework that jointly optimizes retrieval configurations and pricing. Stage I employs contextual clustering with confidence-based exploration to learn effective configurations with logarithmic regret. Stage II adopts interval-based price selection with local Taylor approximation to estimate buyer responses and achieve sublinear regret. We establish theoretical guarantees with polynomial time complexity and validate the framework on four real-world datasets, demonstrating consistent improvements in cumulative reward and regret reduction compared with existing methods.

Explicit latent variable models provide a flexible yet powerful framework for data synthesis, enabling controlled manipulation of generative factors. With latent variables drawn from a tractable probability density function that can be further constrained, these models enable continuous and semantically rich exploration of the output space by navigating their latent spaces. Structured latent representations are typically obtained through the joint minimization of regularization loss functions. In variational information bottleneck models, reconstruction loss and Kullback-Leibler Divergence (KLD) are often linearly combined with an auxiliary Attribute-Regularization (AR) loss. However, balancing KLD and AR turns out to be a very delicate matter. When KLD dominates over AR, generative models tend to lack controllability; when AR dominates over KLD, the stochastic encoder is encouraged to violate the standard normal prior. We explore this trade-off in the context of symbolic music generation with explicit control over continuous musical attributes. We show that existing approaches struggle to jointly minimize both regularization objectives, whereas suitable attribute transformations can help achieve both controllability and regularization of the target latent dimensions.

Bayesian inference, while foundational to probabilistic reasoning, is often hampered by the computational intractability of posterior distributions, particularly through the challenging evidence integral. Conventional approaches like Markov Chain Monte Carlo (MCMC) and Variational Inference (VI) face significant scalability and efficiency limitations. This paper introduces a novel, unifying framework for fast Bayesian updates by leveraging harmonic analysis. We demonstrate that representing the prior and likelihood in a suitable orthogonal basis transforms the Bayesian update rule into a spectral convolution. Specifically, the Fourier coefficients of the posterior are shown to be the normalized convolution of the prior and likelihood coefficients. To achieve computational feasibility, we introduce a spectral truncation scheme, which, for smooth functions, yields an exceptionally accurate finite-dimensional approximation and reduces the update to a circular convolution. This formulation allows us to exploit the Fast Fourier Transform (FFT), resulting in a deterministic algorithm with O(N log N) complexity -- a substantial improvement over the O(N^2) cost of naive methods. We establish rigorous mathematical criteria for the applicability of our method, linking its efficiency to the smoothness and spectral decay of the involved distributions. The presented work offers a paradigm shift, connecting Bayesian computation to signal processing and opening avenues for real-time, sequential inference in a wide class of problems.

Large Language Models (LLMs) have demonstrated remarkable capabilities across numerous tasks, yet principled explanations for their underlying mechanisms and several phenomena, such as scaling laws, hallucinations, and related behaviors, remain elusive. In this work, we revisit the classical relationship between compression and prediction, grounded in Kolmogorov complexity and Shannon information theory, to provide deeper insights into LLM behaviors. By leveraging the Kolmogorov Structure Function and interpreting LLM compression as a two-part coding process, we offer a detailed view of how LLMs acquire and store information across increasing model and data scales -- from pervasive syntactic patterns to progressively rarer knowledge elements. Motivated by this theoretical perspective and natural assumptions inspired by Heap's and Zipf's laws, we introduce a simplified yet representative hierarchical data-generation framework called the Syntax-Knowledge model. Under the Bayesian setting, we show that prediction and compression within this model naturally lead to diverse learning and scaling behaviors observed in LLMs. In particular, our theoretical analysis offers intuitive and principled explanations for both data and model scaling laws, the dynamics of knowledge acquisition during training and fine-tuning, factual knowledge hallucinations in LLMs. The experimental results validate our theoretical predictions.

This work introduces the quantum-inspired variational convolution (QiVC) framework, a novel learning paradigm that integrates principles of probabilistic inference, variational optimization, and quantum-inspired transformations within convolutional architectures. The central innovation of QiVC lies in its quantum-inspired rotated ensemble (QiRE) mechanism. QiRE performs differentiable low-dimensional subspace rotations of convolutional weights, analogously to quantum state evolution. This approach enables structured uncertainty modeling while preserving the intrinsic geometry of the parameter space, resulting in more expressive, stable, and uncertainty-aware representations. To demonstrate its practical potential, the concept is instantiated in a QiVC-based convolutional network (QiVC-Net) and evaluated in the context of biosignal classification, focusing on phonocardiogram (PCG) recordings, a challenging domain characterized by high noise, inter-subject variability, and often imbalanced data. The proposed QiVC-Net integrates an architecture in which the QiVC layer does not introduce additional parameters, instead performing an ensemble rotation of the convolutional weights through a structured mechanism ensuring robustness without added highly computational burden. Experiments on two benchmark datasets, PhysioNet CinC 2016 and PhysioNet CirCor DigiScope 2022, show that QiVC-Net achieves state-of-the-art performance, reaching accuracies of 97.84% and 97.89%, respectively. These findings highlight the versatility of the QiVC framework and its promise for advancing uncertainty-aware modeling in real-world biomedical signal analysis. The implementation of the QiVConv layer is openly available in GitHub.

Ordinal categorical data are routinely encountered in a wide range of practical applications. When the primary goal is to construct a regression model for ordinal outcomes, cumulative link models represent one of the most popular choices to link the cumulative probabilities of the response with a set of covariates through a parsimonious linear predictor, shared across response categories. When the number of observations grows, standard sampling algorithms for Bayesian inference scale poorly, making posterior computation increasingly challenging in large datasets. In this article, we propose three scalable algorithms for approximating the posterior distribution of the regression coefficients in cumulative probit models relying on Variational Bayes and Expectation Propagation. We compare the proposed approaches with inference based on Markov Chain Monte Carlo, demonstrating superior computational performance and remarkable accuracy; finally, we illustrate the utility of the proposed algorithms on a challenging case study to investigate the structure of a criminal network.

The quantity of interest in the classical Cramér-Rao theory of unbiased estimation (e.g., the Cramér-Rao lower bound, its exact attainment for exponential families, and asymptotic efficiency of maximum likelihood estimation) is the variance, which represents the instability of an estimator when its value is compared to the value for an independently-sampled data set from the same distribution. In this paper we are interested in a quantity which represents the instability of an estimator when its value is compared to the value for an infinitesimal additive perturbation of the original data set; we refer to this as the "sensitivity" of an estimator. The resulting theory of sensitivity is based on the Wasserstein geometry in the same way that the classical theory of variance is based on the Fisher-Rao (equivalently, Hellinger) geometry, and this insight allows us to determine a collection of results which are analogous to the classical case: a Wasserstein-Cramér-Rao lower bound for the sensitivity of any unbiased estimator, a characterization of models in which there exist unbiased estimators achieving the lower bound exactly, and some concrete results that show that the Wasserstein projection estimator achieves the lower bound asymptotically. We use these results to treat many statistical examples, sometimes revealing new optimality properties for existing estimators and other times revealing entirely new estimators.