We introduce a novel framework for differentially private (DP) statistical estimation via data truncation, addressing a key challenge in DP estimation when the data support is unbounded. Traditional approaches rely on problem-specific sensitivity analysis, limiting their applicability. By leveraging techniques from truncated statistics, we develop computationally efficient DP estimators for exponential family distributions, including Gaussian mean and covariance estimation, achieving near-optimal sample complexity. Previous works on exponential families only consider bounded or one-dimensional families. Our approach mitigates sensitivity through truncation while carefully correcting for the introduced bias using maximum likelihood estimation and DP stochastic gradient descent. Along the way, we establish improved uniform convergence guarantees for the log-likelihood function of exponential families, which may be of independent interest. Our results provide a general blueprint for DP algorithm design via truncated statistics.

This paper extends Bayesian probability theory by developing a multidimensional space of events (MDSE) theory that accounts for mutual influences between events and hypotheses sets. While traditional Bayesian approaches assume conditional independence between certain variables, real-world systems often exhibit complex interdependencies that limit classical model applicability. Building on established probabilistic foundations, our approach introduces a mathematical formalism for modeling these complex relationships. We developed the MDSE theory through rigorous mathematical derivation and validated it using three complementary methodologies: analytical proofs, computational simulations, and case studies drawn from diverse domains. Results demonstrate that MDSE successfully models complex dependencies with 15-20% improved prediction accuracy compared to standard Bayesian methods when applied to datasets with high interdimensionality. This theory particularly excels in scenarios with over 50 interrelated variables, where traditional methods show exponential computational complexity growth while MDSE maintains polynomial scaling. Our findings indicate that MDSE provides a viable mathematical foundation for extending Bayesian reasoning to complex systems while maintaining computational tractability. This approach offers practical applications in engineering challenges including risk assessment, resource optimization, and forecasting problems where multiple interdependent factors must be simultaneously considered.

Sequential Monte Carlo (SMC) methods offer a principled approach to Bayesian uncertainty quantification but are traditionally limited by the need for full-batch gradient evaluations. We introduce a scalable variant by incorporating Stochastic Gradient Hamiltonian Monte Carlo (SGHMC) proposals into SMC, enabling efficient mini-batch based sampling. Our resulting SMCSGHMC algorithm outperforms standard stochastic gradient descent (SGD) and deep ensembles across image classification, out-of-distribution (OOD) detection, and transfer learning tasks. We further show that SMCSGHMC mitigates overfitting and improves calibration, providing a flexible, scalable pathway for converting pretrained neural networks into well-calibrated Bayesian models.

Gaussian Process based Bayesian Optimization is a widely applied algorithm to learn and optimize under uncertainty, well-known for its sample efficiency. However, recently -- and more frequently -- research studies have empirically demonstrated that the Gaussian Process fitting procedure at its core could be its most relevant weakness. Fitting a Gaussian Process means tuning its kernel's hyperparameters to a set of observations, but the common Maximum Likelihood Estimation technique, usually appropriate for learning tasks, has shown different criticalities in Bayesian Optimization, making theoretical analysis of this algorithm an open challenge. Exploiting the analogy between Gaussian Processes and Gaussian Distributions, we present a new approach which uses a prefixed set of hyperparameters values to fit as many Gaussian Processes and then combines them into a unique model as a Wasserstein Barycenter of Gaussian Processes. We considered both "easy" test problems and others known to undermine the \textit{vanilla} Bayesian Optimization algorithm. The new method, namely Wasserstein Barycenter Gausssian Process based Bayesian Optimization (WBGP-BO), resulted promising and able to converge to the optimum, contrary to vanilla Bayesian Optimization, also on the most "tricky" test problems.

This paper introduces Attribution Projection Calculus (AP-Calculus), a novel mathematical framework for determining causal relationships in structured Bayesian networks. We investigate a specific network architecture with source nodes connected to destination nodes through intermediate nodes, where each input maps to a single label with maximum marginal probability. We prove that for each label, exactly one intermediate node acts as a deconfounder while others serve as confounders, enabling optimal attribution of features to their corresponding labels. The framework formalizes the dual nature of intermediate nodes as both confounders and deconfounders depending on the context, and establishes separation functions that maximize distinctions between intermediate representations. We demonstrate that the proposed network architecture is optimal for causal inference compared to alternative structures, including those based on Pearl's causal framework. AP-Calculus provides a comprehensive mathematical foundation for analyzing feature-label attributions, managing spurious correlations, quantifying information gain, ensuring fairness, and evaluating uncertainty in prediction models, including large language models. Theoretical verification shows that AP-Calculus not only extends but can also subsume traditional do-calculus for many practical applications, offering a more direct approach to causal inference in supervised learning contexts.

Over the past decades, there has been a surge of interest in studying low-dimensional structures within high-dimensional data. Statistical factor models $-$ i.e., low-rank plus diagonal covariance structures $-$ offer a powerful framework for modeling such structures. However, traditional methods for fitting statistical factor models, such as principal component analysis (PCA) or maximum likelihood estimation assuming the data is Gaussian, are highly sensitive to heavy tails and outliers in the observed data. In this paper, we propose a novel expectation-maximization (EM) algorithm for robustly fitting statistical factor models. Our approach is based on Tyler's M-estimator of the scatter matrix for an elliptical distribution, and consists of solving Tyler's maximum likelihood estimation problem while imposing a structural constraint that enforces the low-rank plus diagonal covariance structure. We present numerical experiments on both synthetic and real examples, demonstrating the robustness of our method for direction-of-arrival estimation in nonuniform noise and subspace recovery.

Sparse mixture of experts (SMoE) offers an appealing solution to scale up the model complexity beyond the mean of increasing the network's depth or width. However, we argue that effective SMoE training remains challenging because of the suboptimal routing process where experts that perform computation do not directly contribute to the routing process. In this work, we propose competition, a novel mechanism to route tokens to experts with the highest neural response. Theoretically, we show that the competition mechanism enjoys a better sample efficiency than the traditional softmax routing. Furthermore, we develop CompeteSMoE, a simple yet effective algorithm to train large language models by deploying a router to learn the competition policy, thus enjoying strong performances at a low training overhead. Our extensive empirical evaluations on both the visual instruction tuning and language pre-training tasks demonstrate the efficacy, robustness, and scalability of CompeteSMoE compared to state-of-the-art SMoE strategies. We have made the implementation available at: https://github.com/Fsoft-AIC/CompeteSMoE. This work is an improved version of the previous study at arXiv:2402.02526

Photoplethysmography (PPG) signals encode information about relative changes in blood volume that can be used to assess various aspects of cardiac health non-invasively, e.g.\ to detect atrial fibrillation (AF) or predict blood pressure (BP). Deep networks are well-equipped to handle the large quantities of data acquired from wearable measurement devices. However, they lack interpretability and are prone to overfitting, leaving considerable risk for poor performance on unseen data and misdiagnosis. Here, we describe the use of two scalable uncertainty quantification techniques: Monte Carlo Dropout and the recently proposed Improved Variational Online Newton. These techniques are used to assess the trustworthiness of models trained to perform AF classification and BP regression from raw PPG time series. We find that the choice of hyperparameters has a considerable effect on the predictive performance of the models and on the quality and composition of predicted uncertainties. E.g. the stochasticity of the model parameter sampling determines the proportion of the total uncertainty that is aleatoric, and has varying effects on predictive performance and calibration quality dependent on the chosen uncertainty quantification technique and the chosen expression of uncertainty. We find significant discrepancy in the quality of uncertainties over the predicted classes, emphasising the need for a thorough evaluation protocol that assesses local and adaptive calibration. This work suggests that the choice of hyperparameters must be carefully tuned to balance predictive performance and calibration quality, and that the optimal parameterisation may vary depending on the chosen expression of uncertainty.

Classifiers are often deployed in contexts in which the independent and identically distributed (IID) assumption is violated, i.e., in which the data used to train the model and the future data to be classified are not drawn from the same distribution. This situation is generally referred to as dataset shift in the machine learning literature [Storkey, 2009]. In this context, three problems have gained increased attention in the last years. Classifier calibration [Flach and Webb, 2016, Silva Filho et al., 2023] concerns the manipulation of the confidence scores produced by a classifier so that these effectively reflect the likelihood that a given instance is positive. Quantification [Gonz alez et al., 2017, Esuli et al., 2023] is instead concerned with estimating the prevalence of the classes of interest in an unlabelled set. Finally, classifier accuracy prediction aims at inferring how well a classifier will fare on unseen data [Elsahar and Gall e, 2019, Guillory et al., 2021]. Well-established procedures for attaining these three goals when the IID assumption holds are known and routinely used. For instance, calibrating the classifier's outputs can be attained by learning a calibration map (a function mapping classifier confidence scores into values reflecting the likelihood of the positive class) on held-out validation data [Platt, 2000, Zadrozny and Elkan, 2001a, Barlow and Brunk, 1972].

We introduce NeuralSurv, the first deep survival model to incorporate Bayesian uncertainty quantification. Our non-parametric, architecture-agnostic framework flexibly captures time-varying covariate-risk relationships in continuous time via a novel two-stage data-augmentation scheme, for which we establish theoretical guarantees. For efficient posterior inference, we introduce a mean-field variational algorithm with coordinate-ascent updates that scale linearly in model size. By locally linearizing the Bayesian neural network, we obtain full conjugacy and derive all coordinate updates in closed form. In experiments, NeuralSurv delivers superior calibration compared to state-of-the-art deep survival models, while matching or exceeding their discriminative performance across both synthetic benchmarks and real-world datasets. Our results demonstrate the value of Bayesian principles in data-scarce regimes by enhancing model calibration and providing robust, well-calibrated uncertainty estimates for the survival function.