Group structures are common in regression analysis. They can appear in the form of categorical predictors represented by groups of dummy variables or in the context of additive modeling, where each predictor can be expressed as a set of basis functions forming a group; in applications such as gene expression analysis and financial market modeling, groupings exist naturally in the data. For instance, genes that influence similar traits form groups in gene expression data, while stocks from the same sector form groups in financial data. In these scenarios, group shrinkage plays an important role: when there is insufficient evidence to suggest the significance of predictors within a group, the entire group of predictors is shrunk towards zero. This reduces the noise from individual "spurious predictors", which tend to appear more frequently in high-dimensional settings, and decreases model complexity, thereby reducing the risk of overfitting. 1 Within the Bayesian framework, there has been extensive research focusing on the application of continuous shrinkage priors for linear regression problems involving group predictor variables. Traditional approaches, such as the group lasso[31, 24], the group bridge [16], and the group horseshoe [29] primarily apply shrinkage at the group level and do not consider within-group shrinkage.

Neural estimators are simulation-based estimators for the parameters of a family of statistical models, which build a direct mapping from the sample to the parameter vector. They benefit from the versatility of available network architectures and efficient training methods developed in the field of deep learning. Neural estimators are amortized in the sense that, once trained, they can be applied to any new data set with almost no computational cost. While many papers have shown very good performance of these methods in simulation studies and real-world applications, so far no statistical guarantees are available to support these observations theoretically. In this work, we study the risk of neural estimators by decomposing it into several terms that can be analyzed separately. We formulate easy-to-check assumptions ensuring that each term converges to zero, and we verify them for popular applications of neural estimators. Our results provide a general recipe to derive theoretical guarantees also for broader classes of architectures and estimation problems.

Band-limited functions are fundamental objects that are widely used in systems theory and signal processing. In this paper we refine a recent nonparametric, nonasymptotic method for constructing simultaneous confidence regions for band-limited functions from noisy input-output measurements, by working in a Paley-Wiener reproducing kernel Hilbert space. Kernel norm bounds are tightened using a uniformly-randomized Hoeffding's inequality for small samples and an empirical Bernstein bound for larger ones. We derive an approximate threshold, based on the sample size and how informative the inputs are, that governs which bound to deploy. Finally, we apply majority voting to aggregate confidence sets from random subsamples, boosting both stability and region size. We prove that even per-input aggregated intervals retain their simultaneous coverage guarantee. These refinements are also validated through numerical experiments.

We propose a novel nonparametric regression method that models complex input-output relationships as the sum of convex and concave components. The method-Identifiable Convex-Concave Nonparametric Least Squares (ICCNLS)-decomposes the target function into additive shape-constrained components, each represented via sub-gradient-constrained affine functions. To address the affine ambiguity inherent in convex-concave decompositions, we introduce global statistical orthogonality constraints, ensuring that residuals are uncorrelated with both intercept and input variables. This enforces decomposition identifiability and improves interpretability. We further incorporate L1, L2 and elastic net regularisation on sub-gradients to enhance generalisation and promote structural sparsity. The proposed method is evaluated on synthetic and real-world datasets, including healthcare pricing data, and demonstrates improved predictive accuracy and model simplicity compared to conventional CNLS and difference-of-convex (DC) regression approaches. Our results show that statistical identifiability, when paired with convex-concave structure and sub-gradient regularisation, yields interpretable models suited for forecasting, benchmarking, and policy evaluation.

Parameter estimation is a foundational step in statistical modeling, enabling us to extract knowledge from data and apply it effectively. Bayesian estimation of parameters incorporates prior beliefs with observed data to infer distribution parameters probabilistically and robustly. Moreover, it provides full posterior distributions, allowing uncertainty quantification and regularization, especially useful in small or truncated samples. Utilizing the left-truncated log-logistic (LTLL) distribution is particularly well-suited for modeling time-to-event data where observations are subject to a known lower bound such as precipitation data and cancer survival times. In this paper, we propose a Bayesian approach for estimating the parameters of the LTLL distribution with a fixed truncation point \( x_L > 0 \). Given a random variable \( X \sim LL(α, β; x_L) \), where \( α> 0 \) is the scale parameter and \( β> 0 \) is the shape parameter, the likelihood function is derived based on a truncated sample \( X_1, X_2, \dots, X_N \) with \( X_i > x_L \). We assume independent prior distributions for the parameters, and the posterior inference is conducted via Markov Chain Monte Carlo sampling, specifically using the Metropolis-Hastings algorithm to obtain posterior estimates \( \hatα \) and \( \hatβ \). Through simulation studies and real-world applications, we demonstrate that Bayesian estimation provides more stable and reliable parameter estimates, particularly when the likelihood surface is irregular due to left truncation. The results highlight the advantages of Bayesian inference outperform the estimation of parameter uncertainty in truncated distributions for time to event data analysis.

Graph generation has emerged as a critical task in fields ranging from molecule design to drug discovery. Contemporary approaches, notably diffusion and flow-based models, have achieved solid graph generative performance through constructing a probability path that interpolates between a reference distribution and the data distribution. However, these methods typically model the evolution of individual nodes and edges independently and use linear interpolations to build the path assuming that the data lie in Euclidean space. We show that this is suboptimal given the intrinsic non-Euclidean structure and interconnected patterns of graphs, and it poses risks to the sampling convergence. To build a better probability path, we model the joint evolution of the nodes and edges by representing graphs as connected systems parameterized by Markov random fields (MRF). We then leverage the optimal transport displacement between MRF objects to design the probability path for graph generation. Based on this, we introduce BWFlow, a flow-matching framework for graph generation that respects the underlying geometry of graphs and provides smooth velocities in the probability path. The novel framework can be adapted to both continuous and discrete flow-matching algorithms. Experimental evaluations in plain graph generation and 2D/3D molecule generation validate the effectiveness of BWFlow in graph generation with competitive performance, stable training, and guaranteed sampling convergence.

This paper focuses on the design of spatial experiments to optimize the amount of information derived from the experimental data and enhance the accuracy of the resulting causal effect estimator. We propose a surrogate function for the mean squared error (MSE) of the estimator, which facilitates the use of classical graph cut algorithms to learn the optimal design. Our proposal offers three key advances: (1) it accommodates moderate to large spatial interference effects; (2) it adapts to different spatial covariance functions; (3) it is computationally efficient. Theoretical results and numerical experiments based on synthetic environments and a dispatch simulator that models a city-scale ridesharing market, further validate the effectiveness of our design. A python implementation of our method is available at https://github.

Robust distributed learning algorithms aim to maintain good performance in distributed and federated settings, even in the presence of misbehaving workers. Two primary threat models have been studied: Byzantine attacks, where misbehaving workers can send arbitrarily corrupted updates, and data poisoning attacks, where misbehavior is limited to manipulation of local training data. While prior work has shown comparable optimization error under both threat models, a fundamental question remains open: How do these threat models impact generalization? Empirical evidence suggests a gap between the two threat models, yet it remains unclear whether it is fundamental or merely an artifact of suboptimal attacks. In this work, we present the first theoretical investigation into this problem, formally showing that Byzantine attacks are intrinsically more harmful to generalization than data poisoning. Specifically, we prove that: (i) under data poisoning, the uniform algorithmic stability of a robust distributed learning algorithm, with optimal optimization error, degrades by an additive factor of $\varTheta ( \frac{f}{n-f} )$, with $f$ the number of misbehaving workers out of $n$; and (ii) In contrast, under Byzantine attacks, the degradation is in $\mathcal{O} \big( \sqrt{ \frac{f}{n-2f}} \big)$.This difference in stability leads to a generalization error gap that is especially significant as $f$ approaches its maximum value $\frac{n}{2}$.

--Probabilistic models are often used to make predictions in regions of the data space where no observations are available, but it is not always clear whether such predictions are well-informed by previously seen data. In this paper, we propose a knowledge score for predictions from Gaussian process regression (GPR) models that quantifies the extent to which observing data have reduced our uncertainty about a prediction. The knowledge score is interpretable and naturally bounded between 0 and 1. We demonstrate in several experiments that the knowledge score can anticipate when predictions from a GPR model are accurate, and that this anticipation improves performance in tasks such as anomaly detection, extrapolation, and missing data imputation. Index T erms --anomaly detection, Gaussian processes, regression models, trustworthy machine learning, predictive distributions. The task of prediction is of fundamental importance in many domains.

Progress of AI has led to a creation of very successful, but by no means humble models and tools, especially regarding (i) the huge and further exploding costs and resources they demand, and (ii) the over-confidence of these tools with the answers they provide. Here we introduce a novel mathematical framework for a non-equilibrium entropy-optimizing reformulation of Boltzmann machines based on the exact law of total probability. It results in the highly-performant, but much cheaper, gradient-descent-free learning framework with mathematically-justified existence and uniqueness criteria, and answer confidence/reliability measures. Comparisons to state-of-the-art AI tools in terms of performance, cost and the model descriptor lengths on a set of synthetic problems with varying complexity reveal that the proposed method results in more performant and slim models, with the descriptor lengths being very close to the intrinsic complexity scaling bounds for the underlying problems. Applying this framework to historical climate data results in models with systematically higher prediction skills for the onsets of La Niña and El Niño climate phenomena, requiring just few years of climate data for training - a small fraction of what is necessary for contemporary climate prediction tools.