Quantifying model uncertainty is critical for understanding prediction reliability, yet distinguishing between aleatoric and epistemic uncertainty remains challenging. We extend recent work from classification to regression to provide a novel frequentist approach to epistemic and aleatoric uncertainty estimation. We train models to generate conditional predictions by feeding their initial output back as an additional input. This method allows for a rigorous measurement of model uncertainty by observing how prediction responses change when conditioned on the model's previous answer. We provide a complete theoretical framework to analyze epistemic uncertainty in regression in a frequentist way, and explain how it can be exploited in practice to gauge a model's uncertainty, with minimal changes to the original architecture.

An important goal in cancer research is the survival prognosis of a patient based on a minimal panel of genomic and molecular markers such as genes or proteins. Purely data-driven models without any biological knowledge can produce non-interpretable results. We propose a penalized semiparametric Bayesian Cox model with graph-structured selection priors for sparse identification of multi-omics features by making use of a biologically meaningful graph via a Markov random field (MRF) prior to capturing known relationships between multi-omics features. Since the fixed graph in the MRF prior is for the prior probability distribution, it is not a hard constraint to determine variable selection, so the proposed model can verify known information and has the potential to identify new and novel biomarkers for drawing new biological knowledge. Our simulation results show that the proposed Bayesian Cox model with graph-based prior knowledge results in more trustable and stable variable selection and non-inferior survival prediction, compared to methods modeling the covariates independently without any prior knowledge. The results also indicate that the performance of the proposed model is robust to a partially correct graph in the MRF prior, meaning that in a real setting where not all the true network information between covariates is known, the graph can still be useful. The proposed model is applied to the primary invasive breast cancer patients data in The Cancer Genome Atlas project.

Effective uncertainty quantification is important for training modern predictive models with limited data, enhancing both accuracy and robustness. While Bayesian methods are effective for this purpose, they can be challenging to scale. When employing approximate Bayesian inference, ensuring the quality of samples from the posterior distribution in a computationally efficient manner is essential. This paper addresses the estimation of the Bayesian posterior to generate diverse samples by approximating the gradient flow of the Kullback-Leibler (KL) divergence and the cross entropy of the target approximation under the metric induced by the Stein Operator. It presents empirical evaluations on classification tasks to assess the method's performance and discuss its effectiveness for Model-Based Reinforcement Learning that uses uncertainty-aware network dynamics models.

In this manuscript we introduce a novel Decision Flow (DF) framework for sampling from a target distribution while incorporating additional guidance from a prior sampler. DF can be viewed as an AI driven algorithmic reincarnation of the Markov Decision Process (MDP) approach in Stochastic Optimal Control. It extends the continuous space, continuous time path Integral Diffusion sampling technique to discrete time and space, while also generalizing the Generative Flow Network framework. In its most basic form, an explicit, Neural Network (NN) free formulation, DF leverages the linear solvability of the the underlying MDP to adjust the transition probabilities of the prior sampler. The resulting Markov Process is expressed as a convolution of the reverse time Green's function of the prior sampling with the target distribution. We illustrate the DF framework through an example of sampling from the Ising model, discuss potential NN based extensions, and outline how DF can enhance guided sampling across various applications.

This paper introduces the Heteroscedastic AddiVortes model, a Bayesian non-parametric regression framework that simultaneously models the conditional mean and variance of a response variable using adaptive Voronoi tessellations. By employing a sum-of-tessellations approach for the mean and a product-of-tessellations approach for the variance, the model provides a flexible and interpretable means to capture complex, predictor-dependent relationships and heteroscedastic patterns in data. This dual-layer representation enables precise inference, even in high-dimensional settings, while maintaining computational feasibility through efficient Markov Chain Monte Carlo (MCMC) sampling and conjugate prior structures. We illustrate the model's capability through both simulated and real-world datasets, demonstrating its ability to capture nuanced variance structures, provide reliable predictive uncertainty quantification, and highlight key predictors influencing both the mean response and its variability. Empirical results show that the Heteroscedastic AddiVortes model offers a substantial improvement in capturing distributional properties compared to both homoscedastic and heteroscedastic alternatives, making it a robust tool for complex regression problems in various applied settings.

Suppose we observe a trajectory of length $n$ from an $\alpha$-mixing stochastic process over a finite but potentially large state space. We consider the problem of estimating the probability mass placed by the stationary distribution of any such process on elements that occur with a certain frequency in the observed sequence. We estimate this vector of probabilities in total variation distance, showing universal consistency in $n$ and recovering known results for i.i.d. sequences as special cases. Our proposed methodology carefully combines the plug-in (or empirical) estimator with a recently-proposed modification of the Good--Turing estimator called WingIt, which was originally developed for Markovian sequences. En route to controlling the error of our estimator, we develop new performance bounds on WingIt and the plug-in estimator for $\alpha$-mixing stochastic processes. Importantly, the extensively used method of Poissonization can no longer be applied in our non i.i.d. setting, and so we develop complementary tools -- including concentration inequalities for a natural self-normalized statistic of mixing sequences -- that may prove independently useful in the design and analysis of estimators for related problems.

In real-life scenarios, a Reinforcement Learning (RL) agent aiming to maximise their reward, must often also behave in a safe manner, including at training time. Thus, much attention in recent years has been given to Safe RL, where an agent aims to learn an optimal policy among all policies that satisfy a given safety constraint. However, strict safety guarantees are often provided through approaches based on linear programming, and thus have limited scaling. In this paper we present a new, scalable method, which enjoys strict formal guarantees for Safe RL, in the case where the safety dynamics of the Markov Decision Process (MDP) are known, and safety is defined as an undiscounted probabilistic avoidance property. Our approach is based on state-augmentation of the MDP, and on the design of a shield that restricts the actions available to the agent. We show that our approach provides a strict formal safety guarantee that the agent stays safe at training and test time. Furthermore, we demonstrate that our approach is viable in practice through experimental evaluation.

In this paper, we present the amortized optimal transport filter (A-OTF) designed to mitigate the computational burden associated with the real-time training of optimal transport filters (OTFs). OTFs can perform accurate non-Gaussian Bayesian updates in the filtering procedure, but they require training at every time step, which makes them expensive. The proposed A-OTF framework exploits the similarity between OTF maps during an initial/offline training stage in order to reduce the cost of inference during online calculations. More precisely, we use clustering algorithms to select relevant subsets of pre-trained maps whose weighted average is used to compute the A-OTF model akin to a mixture of experts. A series of numerical experiments validate that A-OTF achieves substantial computational savings during online inference while preserving the inherent flexibility and accuracy of OTF.

Effectively understanding and managing CVD requires advanced diagnostic tools capable of accurately characterizing complex hemodynamics within the cardiovascular system. While medical imaging modalities such as computed tomography (CT) and magnetic resonance imaging (MRI) provide high-resolution anatomical detail, they lack the capability to directly capture hemodynamics information (e.g., blood flow patterns, pressure, and wall shear stress fields) critical for understanding vascular function and pathology. To bridge this gap, image-based computational fluid dynamics (CFD) has emerged as a powerful computational paradigm that derives hemodynamic information from anatomical images via conservation laws. Although widely utilized in cardiovascular research, the clinical application of image-based CFD for diagnosis and surgical planning remains limited, largely due to the challenges associated with efficient and accurate model construction [2-4]. Constructing patient-specific vascular models for image-based CFD involves multiple steps, including image segmentation, geometry modeling, and mesh generation for the computational domain, all of which are critical to ensuring the fidelity of the final simulation results. However, the standard workflow heavily relies on manual methods, making it highly labor-intensive and time-consuming.

Ensuring safe interactions between autonomous vehicles (AVs) and human drivers in mixed traffic systems remains a major challenge, particularly in complex, high-risk scenarios. This paper presents a cognition-decision framework that integrates individual variability and commonalities in driver behavior to quantify risk cognition and model dynamic decision-making. First, a risk sensitivity model based on a multivariate Gaussian distribution is developed to characterize individual differences in risk cognition. Then, a cognitive decision-making model based on the drift diffusion model (DDM) is introduced to capture common decision-making mechanisms in highrisk environments. The DDM dynamically adjusts decision thresholds by integrating initial bias, drift rate, and boundary parameters, adapting to variations in speed, relative distance, and risk sensitivity to reflect diverse driving styles and risk preferences. By simulating high-risk scenarios with lateral, longitudinal, and multidimensional risk sources in a driving simulator, the proposed model accurately predicts cognitive responses and decision behaviors during emergency maneuvers. Specifically, by incorporating driver-specific risk sensitivity, the model enables dynamic adjustments of key DDM parameters, allowing for personalized decision-making representations in diverse scenarios. Comparative analysis with IDM, Gipps, and MOBIL demonstrates that DDM more precisely captures human cognitive processes and adaptive decision-making in high-risk scenarios. These findings provide a theoretical basis for modeling human driving behavior and offer critical insights for enhancing AV-human interaction in real-world traffic environments. Introduction Driving safety is directly influenced by drivers' risk cognition and collision avoidance decisionmaking abilities in high-risk scenarios. In real-world driving, risk cognition generally involves complex interactions among multiple co-existing risk factors rather than being limited to a single risk source (Crosato et al., 2024; Huang et al., 2022).