UQ-Robustness: Integrating Uncertainty & Stability

• UQ-Robustness is a framework that jointly analyzes uncertainty quantification (UQ) and robustness quantification (RQ) to evaluate predictive stability under perturbations.
• It employs methods from robust optimization, Bayesian inference, and risk measurement to balance the tradeoffs between model accuracy and epistemic uncertainty.
• Key applications include classifier reliability, distributional perturbations in computer-code UQ, and robust probabilistic prediction in machine learning and risk assessment.

Searching arXiv for papers on UQ robustness, robustness quantification, and related uncertainty-robustness frameworks. UQ-Robustness denotes a line of work in which uncertainty quantification is studied together with robustness to misspecification, ambiguity, perturbations, or inadequate information. In recent research, this conjunction appears in several forms: prediction-level reliability assessment, where uncertainty quantification and robustness quantification are treated as distinct but complementary signals; robustification of computer-code UQ through perturbation classes on input laws; ambiguity-aware risk measurement via uncertainty sets; robust probabilistic prediction for stochastic dynamical systems; and machine-learning methods that seek uncertainty estimates stable to model selection, finite-sample variability, and approximation error (Detavernier et al., 24 Mar 2026, Ketema et al., 2024, Agarwal et al., 13 May 2025, Mimikos-Stamatopoulos et al., 2024).

1. Conceptual foundations

A central conceptual distinction in recent work is between uncertainty and robustness as separate aspects of reliability. In classifier reliability assessment, uncertainty quantification (UQ) is defined as assigning a numerical score intended to quantify or estimate the amount of uncertainty attached to a prediction, whereas robustness quantification (RQ) is defined as assigning a numerical score intended to quantify how much epistemic uncertainty the model could tolerate before the prediction changes. In that formulation, UQ asks “how much uncertainty is there?”, while RQ asks “how much epistemic uncertainty can this prediction withstand without changing?”; the distinction is grounded in the decomposition between aleatoric uncertainty and epistemic uncertainty (Detavernier et al., 24 Mar 2026).

The same literature also argues that the two notions are complementary rather than interchangeable. One study states that there is “no clear winner throughout all the datasets,” but that combining UQ and RQ yields better ranking of reliable versus unreliable predictions, with the learned mixing weight providing a dataset-specific indication of the relative importance of uncertainty and robustness as sources of unreliability (Detavernier et al., 17 Dec 2025). This suggests that UQ-Robustness is not a single metric but a family of designs in which epistemic stability and predictive ambiguity are jointly analyzed.

A different conceptual synthesis appears in “Uncertainty Quantification of the 4th kind,” which introduces a hybrid between robust optimization, Bayesian inference, decision theory, and hypothesis testing. After observing a sample $x$, the method proceeds by defining a likelihood region through the relative likelihood and then playing a minmax game in that region to define optimal estimators and their risk. Its distinguishing claims are that an optimal prior is identified after measuring the data, that the notion of risk is posterior, and that the optimal estimate and its risk can be reduced to computing the minimum enclosing ball of the image of the likelihood region under the quantity-of-interest map. The method is controlled by a parameter in $[0,1]$ acting as a lower bound on the rarity of the observed data, thereby navigating an accuracy-uncertainty tradeoff and a robustness-accuracy tradeoff associated with data assimilation (Bajgiran et al., 2021).

2. Prediction-level reliability and hybrid reliability scores

In prediction-level reliability assessment for probabilistic generative classifiers, the mathematical setup distinguishes $P_{\text{train}}$, $P_{\text{test}}$, and the learned classifier distribution $P_{\text{classif}}$. For an instance with features $f$, the predicted class is

$$\hat c := h(f)=\arg\max_{c\in\mathcal C} P_{\text{classif}}(c\mid f).$$

The UQ scores studied in this setting include the predicted-class error probability

$U_{\max}(f)=1-P_{\text{classif}}(\hat c\mid f),$

the confidence margin

$$U_{\text{conf}}(f)=P_{\text{classif}}(\hat c\mid f)-\max_{c\in\mathcal C\setminus\{\hat c\}}P_{\text{classif}}(c\mid f),$$

and Shannon entropy

$$U_H(f)=-\sum_{c\in\mathcal C} P_{\text{classif}}(c\mid f)\log_2 P_{\text{classif}}(c\mid f).$$

Ensemble-based total, aleatoric, and epistemic uncertainties are also defined through bootstrap-trained models and the decomposition $[0,1]$0 (Detavernier et al., 24 Mar 2026).

The corresponding RQ notions are global and local robustness. Global robustness $[0,1]$1 is based on $[0,1]$2-contamination of the full joint distribution and has the closed form

$[0,1]$3

Local robustness $[0,1]$4 perturbs local model parameters through $[0,1]$5-contamination and is a model-class-specific certified stability radius in parameter space; for Naive Bayes classifiers and Generative Forests, computation typically requires “a combination of optimisation and binary search” (Detavernier et al., 24 Mar 2026). These robustness scores are explicitly distinguished from adversarial robustness in input space.

Evaluation is performed through accuracy-rejection curves (ARCs), in which instances are ordered from least to most reliable and then rejected in that order; the summary statistic is the area under the ARC (AU-ARC), computed as the average of accuracies across all possible rejection rates (Detavernier et al., 24 Mar 2026). On 14 cleaned UCI datasets with discrete features only, local robustness is reported as frequently the best single reliability signal. For Naive Bayes classifiers, $[0,1]$6 wins on 6 of 14 datasets in the standard setting and on 9 of 14 datasets under a high-uncertainty setting with $[0,1]$7 and $[0,1]$8; for Generative Forests, $[0,1]$9 wins on 7 of 14 datasets in the standard setting (Detavernier et al., 24 Mar 2026). A companion study using Naive Bayes classifiers likewise concludes that neither UQ nor RQ dominates across all datasets, and that a rank-based hybrid is usually best or a close second (Detavernier et al., 17 Dec 2025).

The hybrid construction is rank-based rather than score-based: $P_{\text{train}}$0 where $P_{\text{train}}$1 and $P_{\text{train}}$2 are the positions induced by a UQ measure and an RQ measure, respectively. $P_{\text{train}}$3 recovers UQ-only ordering, and $P_{\text{train}}$4 recovers RQ-only ordering. In the broader benchmark study, $P_{\text{train}}$5 is trained on the training set to maximize AU-ARC and then adjusted by a bias parameter $P_{\text{train}}$6 to reduce overfitting (Detavernier et al., 24 Mar 2026). In the Naive Bayes-only study, $P_{\text{train}}$7 is optimized per dataset and per uncertainty/robustness pair, and is interpreted as an operational estimate of the relative importance of uncertainty versus robustness (Detavernier et al., 17 Dec 2025).

3. Distributional perturbations in computer-code UQ

A major strand of UQ-Robustness concerns deterministic computer models $P_{\text{train}}$8 when the input law is uncertain or misspecified. In the Fisher-Rao approach, one specifies for each input $P_{\text{train}}$9 a parametric family $P_{\text{test}}$0 containing the baseline marginal $P_{\text{test}}$1, equips $P_{\text{test}}$2 with the Fisher-Rao metric, and defines perturbation sets as Fisher-Rao spheres

$P_{\text{test}}$3

Robustness is then quantified through perturbed-law indices

$P_{\text{test}}$4

and worst-case bounds

$P_{\text{test}}$5

This framework is justified as being “consistent with the information structure” of the baseline family and is developed in detail for truncated distributions, especially truncated normals and push-forward families such as truncated log-normal laws (Ketema et al., 2024).

A closely related earlier formulation defines perturbations intrinsically on a statistical manifold using the Fisher information metric. For a parametric family $P_{\text{test}}$6, the Fisher information matrix

$P_{\text{test}}$7

induces the Fisher-Rao metric; perturbations of level $P_{\text{test}}$8 are densities at Fisher distance $P_{\text{test}}$9 from the baseline, and the full perturbation class is the Fisher sphere $P_{\text{classif}}$0. Robustness is then measured through quantile-oriented perturbed-law sensitivity indices

$P_{\text{classif}}$1

with extrema

$P_{\text{classif}}$2

This yields robustness envelopes for output quantiles under controlled distributional perturbation of input laws (Gauchy et al., 2020).

Another robust-UQ approach studies worst-case quantiles over moment classes. With

$P_{\text{classif}}$3

and input ambiguity sets defined by support constraints and finitely many moments, the target is

$P_{\text{classif}}$4

Using the OUQ reduction theorem, the extremizing marginals can be chosen as discrete measures with at most $P_{\text{classif}}$5 support points, and canonical moments provide a parameterization in which the free variables lie in a simple box and every parameter vector corresponds to an admissible measure. The resulting method is explicitly distributionally robust and is illustrated on a flood model and a CATHARE thermal-hydraulic case study (Stenger et al., 2018).

4. Robust UQ in machine learning and generative modeling

In machine learning, one robust-UQ program is to account explicitly for model misspecification, model-selection uncertainty, finite-sample variability, algorithmic instability, and subgroup or local miscalibration. PCS-UQ operationalizes this through the Predictability-Computability-Stability framework. Candidate algorithms are screened by a prediction check on validation data; the screened algorithms are then fit across multiple bootstraps, and predictive sets are calibrated by a multiplicative scaling factor $P_{\text{classif}}$6. In regression, the PCS interval has the form

$P_{\text{classif}}$7

while a modified split version is shown to be a form of conformal inference with exchangeable-data coverage. Across 17 regression and 6 classification datasets, PCS-UQ is reported to achieve the desired coverage and reduce width over conformal approaches by approximately $P_{\text{classif}}$8; on three computer-vision benchmarks, approximate PCS-UQ variants based on weighted Monte-Carlo dropout or additive Gaussian weight noise also reduce prediction-set size over conformal methods by $P_{\text{classif}}$9 while being $f$0 to $f$1 faster than the original PCS-UQ (Agarwal et al., 13 May 2025).

QUEST approaches robust regression UQ from a different direction, replacing predictive-risk summaries by highest density regions. For a density $f$2, the $f$3-highest density region is

$f$4

and the local QUEST uncertainty is its Lebesgue volume

$f$5

The global version is

$f$6

Here $f$7 is explicitly interpreted as a robustness parameter controlling the degree of tail area fluctuations deemed practically irrelevant for UQ. The paper states that local and global QUEST measures of aleatoric and epistemic uncertainty satisfy axioms $f$8-$f$9, including monotonicity under spread and invariance to location shifts, and reports that global QUEST outperforms random, variance, entropy, and local QUEST in all selective-prediction trials considered, with the expected advantage strongest in skewed and multimodal settings (Goring et al., 17 Jun 2026).

For score-based generative models, robustness is formulated as model-form UQ. The Wasserstein uncertainty propagation theorem studies two Fokker–Planck equations with drifts $$\hat c := h(f)=\arg\max_{c\in\mathcal C} P_{\text{classif}}(c\mid f).$$0 and $$\hat c := h(f)=\arg\max_{c\in\mathcal C} P_{\text{classif}}(c\mid f).$$1, and turns an $$\hat c := h(f)=\arg\max_{c\in\mathcal C} P_{\text{classif}}(c\mid f).$$2 drift error into a discrepancy bound in Wasserstein-1 and, under stronger assumptions, total variation. In the SGM specialization, $$\hat c := h(f)=\arg\max_{c\in\mathcal C} P_{\text{classif}}(c\mid f).$$3 is the true reverse drift and $$\hat c := h(f)=\arg\max_{c\in\mathcal C} P_{\text{classif}}(c\mid f).$$4 the learned score-based drift, so learning error propagates into a $$\hat c := h(f)=\arg\max_{c\in\mathcal C} P_{\text{classif}}(c\mid f).$$5 ball around the target law. The paper explicitly tracks five error sources—finite sample approximation, early stopping, score-matching objective choice, score-function parametrization expressiveness, and reference-distribution choice—and argues that stochasticity is the key mechanism ensuring SGMs are provably robust (Mimikos-Stamatopoulos et al., 2024).

5. Robust probabilistic prediction and ambiguity-sensitive risk measurement

For stochastic dynamical systems, robust UQ is cast as robust probabilistic prediction rather than robust point estimation. Given a predictive distribution

$$\hat c := h(f)=\arg\max_{c\in\mathcal C} P_{\text{classif}}(c\mid f).$$6

the paper defines the log-likelihood functional

$$\hat c := h(f)=\arg\max_{c\in\mathcal C} P_{\text{classif}}(c\mid f).$$7

and proves

$$\hat c := h(f)=\arg\max_{c\in\mathcal C} P_{\text{classif}}(c\mid f).$$8

so the likelihood functional is a proper scoring rule. A predictor is defined as robust if

$$\hat c := h(f)=\arg\max_{c\in\mathcal C} P_{\text{classif}}(c\mid f).$$9

Under moment-based information sets $U_{\max}(f)=1-P_{\text{classif}}(\hat c\mid f),$0, the paper shows that robust predictors must have polynomial-exponential form; the optimal predictor is uniform under $U_{\max}(f)=1-P_{\text{classif}}(\hat c\mid f),$1, Gaussian under $U_{\max}(f)=1-P_{\text{classif}}(\hat c\mid f),$2, and exponential under $U_{\max}(f)=1-P_{\text{classif}}(\hat c\mid f),$3, with Laplace and Student-$U_{\max}(f)=1-P_{\text{classif}}(\hat c\mid f),$4 constructions proposed as non-polynomial first-order robust predictors. A Kalman-filter-based algorithm is then used to estimate moments and adapt the predictive family online (Xu et al., 2023).

In risk measurement, robustness is treated as sensitivity of law-invariant risk measures to perturbations of the data-generating law. One paper argues that Hampel’s classical notion of qualitative robustness under weak convergence is not suitable for risk measurement because weak convergence ignores tail behavior; it replaces this by $U_{\max}(f)=1-P_{\text{classif}}(\hat c\mid f),$5-weak robustness on Orlicz spaces, proves a Hampel-type characterization in that topology, and defines an index of qualitative robustness

$U_{\max}(f)=1-P_{\text{classif}}(\hat c\mid f),$6

The resulting framework orders risk measures by the weakest integrability assumption under which they remain finite and shows that robustness and sensitivity are traded off continuously rather than dichotomously (Krätschmer et al., 2012).

A more recent extension develops robust quasi-convex risk measures on $U_{\max}(f)=1-P_{\text{classif}}(\hat c\mid f),$7 spaces by attaching to each position $U_{\max}(f)=1-P_{\text{classif}}(\hat c\mid f),$8 an uncertainty set $U_{\max}(f)=1-P_{\text{classif}}(\hat c\mid f),$9 and defining

$$U_{\text{conf}}(f)=P_{\text{classif}}(\hat c\mid f)-\max_{c\in\mathcal C\setminus\{\hat c\}}P_{\text{classif}}(c\mid f),$$0

Two mechanisms yield robust quasi-convexity: either $$U_{\text{conf}}(f)=P_{\text{classif}}(\hat c\mid f)-\max_{c\in\mathcal C\setminus\{\hat c\}}P_{\text{classif}}(c\mid f),$$1 is quasi-convex and $$U_{\text{conf}}(f)=P_{\text{classif}}(\hat c\mid f)-\max_{c\in\mathcal C\setminus\{\hat c\}}P_{\text{classif}}(c\mid f),$$2 is convex, or $$U_{\text{conf}}(f)=P_{\text{classif}}(\hat c\mid f)-\max_{c\in\mathcal C\setminus\{\hat c\}}P_{\text{classif}}(c\mid f),$$3 is quasi-convex or $$U_{\text{conf}}(f)=P_{\text{classif}}(\hat c\mid f)-\max_{c\in\mathcal C\setminus\{\hat c\}}P_{\text{classif}}(c\mid f),$$4-quasi-convex and $$U_{\text{conf}}(f)=P_{\text{classif}}(\hat c\mid f)-\max_{c\in\mathcal C\setminus\{\hat c\}}P_{\text{classif}}(c\mid f),$$5 is merely monotone. Under convex $$U_{\text{conf}}(f)=P_{\text{classif}}(\hat c\mid f)-\max_{c\in\mathcal C\setminus\{\hat c\}}P_{\text{classif}}(c\mid f),$$6 and quasi-convex, cash-subadditive, continuous-from-above $$U_{\text{conf}}(f)=P_{\text{classif}}(\hat c\mid f)-\max_{c\in\mathcal C\setminus\{\hat c\}}P_{\text{classif}}(c\mid f),$$7, the robustified dual representation is

$$U_{\text{conf}}(f)=P_{\text{classif}}(\hat c\mid f)-\max_{c\in\mathcal C\setminus\{\hat c\}}P_{\text{classif}}(c\mid f),$$8

The paper further characterizes robust acceptance families through

$$U_{\text{conf}}(f)=P_{\text{classif}}(\hat c\mid f)-\max_{c\in\mathcal C\setminus\{\hat c\}}P_{\text{classif}}(c\mid f),$$9

and studies how ambiguity inflates capital allocation (Centrone et al., 18 Mar 2026).

6. Recurrent limitations and open directions

The literature is explicit that robustness claims are heavily conditioned by model class, perturbation family, and evaluation protocol. In classifier reliability, the strongest results are currently restricted to probabilistic generative classifiers with discrete features—specifically Naive Bayes classifiers and Generative Forests—and local robustness is model-specific and lacks a universal closed form (Detavernier et al., 24 Mar 2026). In the Naive Bayes-only study, the main experiments assume train and test come from the same underlying distribution, and distribution shift is noted only via prior work (Detavernier et al., 17 Dec 2025).

For geometric perturbation methods in computer-code UQ, deriving or approximating Fisher information matrices and Christoffel symbols, solving geodesic ODEs, and handling possible non-compact Fisher-Rao spheres remain practical bottlenecks; the selection of $$U_H(f)=-\sum_{c\in\mathcal C} P_{\text{classif}}(c\mid f)\log_2 P_{\text{classif}}(c\mid f).$$0 is explicitly left open, and higher-dimensional parameter spaces are identified as challenging (Ketema et al., 2024). Moment-class OUQ, while theoretically finite-dimensional after reduction, still faces the curse of dimensionality because one probability-of-failure evaluation requires $$U_H(f)=-\sum_{c\in\mathcal C} P_{\text{classif}}(c\mid f)\log_2 P_{\text{classif}}(c\mid f).$$1 evaluations of $$U_H(f)=-\sum_{c\in\mathcal C} P_{\text{classif}}(c\mid f)\log_2 P_{\text{classif}}(c\mid f).$$2 (Stenger et al., 2018).

In machine learning, the practical PCS-UQ pipeline and the modified PCS-UQ algorithm are not identical: the finite-sample coverage theorem applies to the modified split version, whereas the main OOB-based pipeline is supported primarily by empirical evidence (Agarwal et al., 13 May 2025). QUEST requires highest-density-region estimation and the paper explicitly notes the computational burden in high dimensions or for non-analytic densities, as well as the dependence of total uncertainty on how $$U_H(f)=-\sum_{c\in\mathcal C} P_{\text{classif}}(c\mid f)\log_2 P_{\text{classif}}(c\mid f).$$3 is approximated in practice (Goring et al., 17 Jun 2026). The score-based generative-model analysis is mathematically strongest in the torus setting and leaves discretization error outside its main theorems (Mimikos-Stamatopoulos et al., 2024).

A consistent cross-paper pattern is that UQ-Robustness is not reducible to a single scalar criterion. In some settings, it is the tradeoff between uncertainty and stability of an individual classification; in others, it is the geometry of admissible perturbations of an input law; elsewhere, it is the tail-sensitive continuity class of a risk functional, the ambiguity set around a position, or the propagation of score-model error into a Wasserstein ball. What unifies these formulations is not a common algorithm but a common requirement: uncertainty estimates are treated as inadequate unless they remain meaningful under the perturbations, misspecifications, and information restrictions that the modeler regards as plausible (Detavernier et al., 24 Mar 2026, Gauchy et al., 2020, Krätschmer et al., 2012, Xu et al., 2023).