Quantifying Risk Under Evolving Uncertainty: Belief-Dependent Robustness for Safe Sequential Decision Making

How cautious should an agent be while it is still learning its environment? We propose RATTL (Risk-Adversarial Total-Reward Learning), which ties caution to epistemic uncertainty: the agent holds a Bayesian posterior over unknown dynamics and plans against a Wasserstein ambiguity set whose radius is a monotone function of that posterior. The radius contracts with evidence, so behaviour interpolates continuously between worst-case robustness and risk-neutral total-reward maximization. The design follows the duality underlying the Entropic Value-at-Risk, which converts the choice of a risk level into the choice of an ambiguity radius. We show the resulting planning problem is well posed under transience and compactness conditions, and prove a Safety Sandwich: the RATTL value lies between the uninformed robust value and the full- knowledge optimum, with a gap that vanishes as the posterior concentrates. In a canonical binary-hazard instance, the induced criterion reduces to Conditional Value-at-Risk at a level set by the posterior entropy. A worked example shows the agent deferring the efficient action until a sharp identification threshold. RATTL targets runtime safety for agents, including LLM-based systems, acting under uncertainty.

Tracing sources of epistemic uncertainty in deep learning predictions: homo- and hetero-scedastic linearized estimators

We adapt two classical statistical estimators for quantifying uncertainty to modern deep learning, in order to provide clearer insights into uncertainty attributable to two sources : aleatoric uncertainty, or locally scarce data. Our approach leverages recent advances in approximate Fisher Information Matrices, to enable scaling to actual architectures. Experimental results demonstrate how each test points is differentially impacted by both sources, highlighting the practical utility of our estimators in improving the robustness of real-world applications.

A Unified Risk View of Uncertainty: Posterior Risk for Disentanglement and Evaluation Beyond Proxies

Reliable uncertainty estimates are critical in safety-sensitive applications, where understanding the sources of predictive uncertainty is essential. This often requires disentangling epistemic uncertainty from aleatoric uncertainty, yet these uncertainty types are not defined consistently across the literature, making it difficult to assess whether a method produces accurate uncertainty estimates. Evaluation is further complicated by the fact that ground-truth epistemic uncertainty is typically unavailable. Existing benchmarks therefore mostly rely on proxy tasks such as out-of-distribution detection, which do not provide complete ground-truth uncertainty targets and offer limited insight into the structure and quality of uncertainty estimates. We propose a unified definition of uncertainty as pointwise posterior risk, the expected loss of a predictor under the distribution of plausible ground-truth functions given the data. This view combines Bayesian uncertainty over functions with estimator-dependent deviations from the posterior mean, capturing effects such as misspecification and optimization error. This formulation constitutes the foundation of a theory-backed benchmark that enables direct computation of oracle epistemic and aleatoric uncertainty using semi-synthetic datasets with real covariates and known generative processes. By avoiding proxy evaluations, the benchmark enables fine-grained analysis of uncertainty estimates. Empirically, we find that accurate prediction does not guarantee reliable uncertainty disentanglement. The benchmark reveals practically useful differences between methods, identifying approaches with meaningful alignment to oracle uncertainty targets while exposing sensitivity to datasets and modeling choices.

From Hybrid Mechanistic--Data-Driven Modeling Toward Neuro-Symbolic AI: What, Why, and How

Hybrid mechanistic/data-driven models, which combine first-principles with learned components, are increasingly used in process engineering and scientific machine learning. Common hybrid modeling designs are specified primarily through their architectures and training losses, which offers a limited basis for a shared semantic interface to compare or verify them across domains, with comparatively little attention paid to epistemic uncertainty in the mechanistic part. We bridge hybrid modeling and neuro-symbolic (NeSy) AI by reconstructing these designs as instances of NeSy interface. The resulting translation, Hybrid-to-NeSy (H2N), places mechanistic knowledge on the language side, learned modules on the belief side, and validity domains together with constraints on the logic side. For each design, H2N then yields an explicit NeSy inference functional and a logic-belief decomposition. From this decomposition we derive two metrics: structural violation rate (SVR), measuring whether the learned belief respects the mechanistic structure; and belief dispersion (BD), measuring how concentrated the learned plausibility is, serving as a hybrid model's epistemic uncertainty in its mechanistic part. We instantiate H2N on a case study of a structured hybrid model for binary classification under label noise and show that models with higher SVR and BD exhibit greater variability in held-out accuracy. Under structural distribution shift, H2N further quantifies a model's uncertainty during extrapolations, whereas test accuracy reveals the same shift only post hoc.

Subjective Risk Decomposition: A New View for Uncertainty Quantification

We present a novel viewpoint for uncertainty quantification. Uncertainty measures are not primitives, in need of axioms and argumentation, but instead consequences, of higher-level modelling decisions. We show how epistemic and aleatoric uncertainty measures can be derived via decomposition of a subjective risk, based on a strictly proper loss. Reverse cross-entropy provides a prominent example, where decomposition recovers the classic information-theoretic uncertainty terms. The same approach recovers numerous measures previously proposed across the UQ literature, providing them a common theoretical foundation. From a practical point of view, this suggests a new approach to UQ: given a modelling scenario and strictly proper loss, the corresponding epistemic and aleatoric terms are induced by the subjective-risk decomposition. We then extend our view to learning theory: we introduce and analyse subjective risk analogues of excess risk, approximation error, and estimation error, and identify the connections to UQ. We consider this a first step towards a full learning-theoretic framework for uncertainty quantification.

Evaluating Epistemic Uncertainty: Beyond OOD Detection and Active Learning

Current evaluation of epistemic uncertainty relies on tasks such as out-ofdistribution detection and active learning. However, the Bayes-optimal decision strategies for these tasks do not coincide with the scores commonly used to quantify epistemic uncertainty. Building on the epistemic reject-option framework, we evaluate epistemic uncertainty using its ability to identify regret, the reducible error. Formulating selective prediction as a constrained optimization over coverage, expected risk, and regret, we prove the optimal selector is a thresholded convex combination of the ground-truth aleatoric and epistemic uncertainties. This theoretical unification exposes a weakness in recent uncertainty disentanglement literature: we demonstrate that standard correlation metrics between learned components do not necessarily predict their actual operational utility. We instead propose to evaluate the achievable risk, regret, coverage surface of the decomposition as a diagnostic for joint disentanglement and utility. Benchmarking standard methods on datasets with dense human annotations reveals that decision-theoretic rankings can disagree substantially with proxy-task rankings, including pairwise rank inversions between methods that are top-ranked on one criterion and bottom-ranked on other.

Heckman-Corrected Epistemic Uncertainty: Selection on Unobservables Defeats Importance Weighting

Training data for machine learning is routinely collected by a selection process the model never sees: loans are observed only when granted, outcomes only when a test was ordered. The standard fixes -- importance weighting, covariate-shift correction, MAR imputation -- assume selection is ignorable given observables. Econometrics solved the harder case in 1979: Heckman's two-equation model jointly fits a probit selection equation and an outcome equation linked through correlated errors, and the inverse-Mills-ratio term corrects for selection on unobservables, where importance weighting is structurally helpless. We instantiate this for deep epistemic uncertainty: a deep outcome network, a linear selection head, and a joint bivariate-normal likelihood over all units, ensembled for predictive variance. In a controlled generator where sampling probability depends on an unobservable correlated (rho up to 0.9) with the outcome noise, deep ensembles, MC dropout, and GP baselines are overconfident exactly where data was avoided: coverage of nominal-90% intervals falls to 64.4% at rho=0.9, and importance weighting with oracle propensities does not fix it (43.1%) -- reweighting corrects the covariate distribution, not the conditional bias E[y|x,selected] != E[y|x]. The Heckman correction restores coverage (88.9%) when the selection equation has an instrument -- a variable affecting selection but not the outcome -- and degrades measurably without one (40.3%); we chart this honesty curve rather than hide it. On real tabular data with induced MNAR selection, the corrected intervals are the best-calibrated (lowest region-ECE) non-oracle method in selected-against regions; baselines matching its raw coverage do so only by over-widening everywhere. Our estimators reproduce classic Stata output to seven digits. We state which identification regime a practitioner is in, and release the code.

Interpretable Uncertainty Routing Separating Emotion Ambiguity from Distribution Shift in Facial Expression Recognition

Facial expression recognition (FER) is inherently ambiguous: human annotators frequently disagree, and models deployed in real environments face distribution shift. Crucially, these two conditions demand different downstream actions, as ambiguous in-distribution faces should be reported with their ambiguity whereas out-of-distribution inputs should be rejected. However, a single uncertainty score conflates the two. In this study, uncertainty decomposition into aleatoric and epistemic components for FER is investigated, and Uncertainty-Aware Routing (UAR), an inference-time routing mechanism that exploits the separation, is introduced. Specifically, aleatoric and epistemic uncertainties are obtained from a Deep Ensemble of fully fine-tuned DINOv2 models and are each validated against an independent external signal: aleatoric against human annotator disagreement, and epistemic against distribution shift induced by image corruptions. The proposed dual-validation protocol reveals that aleatoric recovers annotator disagreement with Spearman correlation 0.66 (95% CI: 0.64-0.68), and epistemic detects corruption-induced shifts, achieving average AUROC of 0.699 at the highest corruption severity. UAR retains approximately 1.8 times more ambiguous in-distribution faces than single-uncertainty routing at a matched out-of-distribution rejection rate. A strong label-distribution-learning baseline achieves comparable disagreement recovery but cannot separate ambiguity from shift and therefore cannot route, establishing that the value of decomposition lies in the separation enabling interpretable and differentiated action selection.

LLM Doesn't Know What It Doesn't Know: Detecting Epistemic Blind Spots via Cross-Model Attribution Divergence on Clinical Tabular Data

Large language models (LLMs) are increasingly applied to structured clinical data, yet whether they can recognize the limits of their own knowledge on such tasks remains unexplored. We study this question through the lens of cross-model attribution divergence with the goal of reducing epistemic uncertainty for structured tasks, comparing Qwen 2.5 7B and XGBoost on a prediction task via attribution divergence analysis. We report four findings. First, LLM verbalized confidence is epistemically vacuous, it outputs a near-constant (0.856-0.937) regardless of whether accuracy is 49% or 75.3%, tracking prompt format rather than prediction quality. Second, the LLM exhibits an inverse difficulty effect: accuracy drops to 64.8% when XGBoost is 99% correct, but matches XGBoost (73.8% vs. 73.1%) when it is moderately uncertain. Third, few-shot examples and SHAP-derived feature evidence are orthogonal, super-additive interventions: they reduce the Attribution Disagreement Score (ADS) from 1.54 to 0.38 and improve accuracy from 49% to 75.3% without training. Fourth, a cross-model calibrator that determined LLM reliability using attribution divergence signals reduces expected calibration error from 0.254 to 0.080, replacing uninformative verbalized confidence with patient-specific reliability estimates, without accessing model internals or requiring repeated inference. We frame these findings as a cold start problem for LLMs on structured data and outline a path toward genuine epistemic self-awareness.

Quantification of Uncertainty with Adversarial Models in Medical Image Segmentation

Reliable pixel-level uncertainty quantification holds the potential to transform clinical workflows by enabling high-fidelity longitudinal monitoring and distinguishing true pathological changes from artifacts. Ideally, these models provide the stability required for critical treatment planning and surgical intervention. However, standard deep learning models often suffer from miscalibration, yielding overconfident predictions that mask underlying vulnerabilities at subtle pathological boundaries. To address this, we propose QUAM-SM, a post-hoc framework using targeted adversarial search to identify "adversarially fragile" pixels. By actively seeking perturbations that expose predictive instability, our method highlights regions where decisions are most vulnerable to being flipped. Importantly, the framework disentangles epistemic uncertainty from aleatoric uncertainty. Experiments on two public datasets with multiple expert annotations demonstrate that QUAM-SM outperforms both standard and recent uncertainty estimation approaches in terms of reliability and boundary sensitivity. Code is available at https://github.com/HanaJebril/quam_sm

Exposure Bias as Epistemic Underidentification in Recursive Forecasting

Recursive multi-step forecasting is usually framed as distribution shift: models are trained on observed histories but deployed on their own predictions. We show this framing is incomplete by proving that, under partial observability or state truncation, recursive rollout is also an epistemic underidentification problem. Even with deterministic latent dynamics, one-step Bayes supervision identifies behavior only on observed contexts and need not identify the deployed recursive predictor once rollout queries self-generated induced states whose correct local targets are not determined by numeric state alone. We formalize this with induced states $Z$ and provenance variables $P$, and derive a decomposition of induced-state error into teacher-forcing/rollout mismatch, representation--class approximation, and provenance information gaps. Empirically, we show that rollout enters a distinct induced-state regime, that fixed induced states define a distinct local corrective task, and that closed-loop gains arise not only from local adaptation but also from changing the induced states visited during rollout. Using a simple binary provenance encoding, provenance-aware correction can further improve performance, though gains are conditional rather than uniform. These results recast exposure bias as reasoning under self-induced epistemic uncertainty.

Epistemic Uncertainty Is Not the Reducible Kind

The standard taxonomy of predictive uncertainty defines epistemic uncertainty as the part removable by collecting more data, while the standard measure identifies it with a mutual-information term. We prove the definition and the measure are extensionally inconsistent. On an explicit construction, the measure assigns all uncertainty to the epistemic class, yet no quantity of training data reduces it. Reducibility is instead a property of the pair (uncertainty, acquisition class), and the dichotomy resolves into three parts: aleatoric, sample-reducible epistemic, and mechanism-reducible epistemic uncertainty. An exact identity for the value of an observation shows that in-distribution data never reduces mechanism-irreducible uncertainty and generically increases it. Ensemble disagreement, the deployed epistemic estimate, tracks the training procedure rather than the epistemic term. It collapses to zero beneath a positive truth under consistent training, and equals hyperparameter-scaled initialization noise under interpolation. A finite-sample falsification test and seed-swept experiments confirm the theory.

What Uncertainties Do We Need for Dynamical Systems?

The distinction between aleatoric and epistemic uncertainty has received considerable attention in machine learning research, mainly in the context of supervised learning but also in other settings such as generative modeling. In this paper, we offer a machine learning perspective on uncertainty modeling for dynamical systems, which has been studied much less so far. In particular, we ask: what uncertainties do we need for dynamical systems? We discuss sources of uncertainty, clarify their nature (aleatoric or epistemic), and consider how the objectives of representing and quantifying uncertainty vary across different tasks.

Can we trust our models? Epistemic calibration in second-order classification

Uncertainty estimation is critical for deploying machine learning models in high-stakes settings. However, classical calibration only assesses the reliability of predicted probabilities and does not evaluate whether epistemic uncertainty estimates are themselves trustworthy. This limitation is particularly relevant for second-order classification models. We introduce epistemic calibration, a principled criterion that measures whether reported epistemic uncertainty faithfully reflects the dispersion of model predictions around the ground truth. We show that epistemic calibration is a strictly stronger notion than classical calibration and captures failure modes invisible to standard metrics. We relate this work to the existing literature through an impossibility theorem that holds under the epistemic calibration hypothesis. To operationalize this concept, we propose the Expected Epistemic Calibration Error (EECE), which we prove to be a consistent estimator of a True Epistemic Calibration Error (TECE). Experiments across a broad range of uncertainty quantification methods show that epistemic calibration is a coherent and meaningful criterion and reveal substantial differences across methods, despite similar predictive performance.

SHAPO: Sharpness-Aware Policy Optimization for Safe Exploration

Safe exploration is a prerequisite for deploying reinforcement learning (RL) agents in safety-critical domains. In this paper, we approach safe exploration through the lens of epistemic uncertainty, where the actor's sensitivity to parameter perturbations serves as a practical proxy for regions of high uncertainty. We propose Sharpness-Aware Policy Optimization (SHAPO), a sharpness-aware policy update rule that evaluates gradients at perturbed parameters, making policy updates pessimistic with respect to the actor's epistemic uncertainty. Analytically we show that this adjustment implicitly reweighs policy gradients, amplifying the influence of rare unsafe actions while tempering contributions from already safe ones, thereby biasing learning toward conservative behavior in under-explored regions. Across several continuous-control tasks, our method consistently improves both safety and task performance over existing baselines, significantly expanding their Pareto frontiers.

Biased Dreams: Limitations to Epistemic Uncertainty Quantification in Latent Space Models

Model-Based Reinforcement Learning distinguishes between physical dynamics models operating on proprioceptive inputs and latent dynamics models operating on high-dimensional image observations. A prominent latent approach is the Recurrent State Space Model used in the Dreamer family. While epistemic uncertainty quantification to inform exploration and mitigate model exploitation is well established for physical dynamics models, its transfer to latent dynamics models has received limited scrutiny. We empirically demonstrate that latent transitions are biased toward well-represented regions of latent space, exhibiting an attractor behavior that can deviate from true environment dynamics. As a result, discrepancies in environment dynamics may not manifest in latent space, undermining the reliability of epistemic uncertainty estimates. Because these attractors often lie in high-reward regions, latent rollouts systematically overestimate predicted rewards. Our findings highlight key limitations of epistemic uncertainty estimation in latent dynamics models and motivate more critical evaluation of this method.

Credal Concept Bottleneck Models for Epistemic-Aleatoric Uncertainty Decomposition

Concept Bottleneck Models (CBMs) predict through human-interpretable concepts, but they typically output point concept probabilities that conflate epistemic uncertainty (reducible model underspecification) with aleatoric uncertainty (irreducible input ambiguity). This makes concept-level uncertainty hard to interpret and, more importantly, hard to act upon. We introduce CREDENCE (Credal Ensemble Concept Estimation), a CBM framework that decomposes concept uncertainty by construction. CREDENCE represents each concept as a credal prediction (a probability interval), derives epistemic uncertainty from disagreement across diverse concept heads, and estimates aleatoric uncertainty via a dedicated ambiguity output trained to match annotator disagreement when available. The resulting signals support prescriptive decisions: automate low-uncertainty cases, prioritize data collection for high-epistemic cases, route high-aleatoric cases to human review, and abstain when both are high. Across several tasks, we show that epistemic uncertainty is positively associated with prediction errors, whereas aleatoric uncertainty closely tracks annotator disagreement, providing guidance beyond error correlation. Our implementation is available at the following link: https://github.com/Tankiit/Credal_Sets/tree/ensemble-credal-cbm