Uncertainty-Aware Sensitivity Analysis

• Uncertainty-aware sensitivity analysis is a framework that explicitly represents model uncertainty, using bounds, distributions, or risk indicators to assess how input variability affects outputs.
• It encompasses methods from local one-at-a-time perturbations to global variance-based techniques, capturing key nonlinearity and interaction effects in complex models.
• The approach enhances decision robustness by mapping uncertainty propagation through calibrated parameters, credal sets, and risk metrics, thereby improving model transparency and reliability.

Searching arXiv for the cited topic and key papers to ground the article. Uncertainty-aware sensitivity analysis is the study of how outputs, inferences, or decisions change when the uncertain elements of a model are represented explicitly rather than fixed at single nominal values. In the cited literature, this explicit representation takes several forms: credal sets for initial and transition probabilities in finite discrete-time Markov chains (Cooman et al., 2014), predictive distributions rather than point predictions in Bayesian supervised learning (Paananen et al., 2019), posterior distributions over calibrated physiological parameters (Thiel et al., 17 Sep 2025), perturbations of input probability measures in uncertainty propagation for differential equations (Ernst et al., 2020), and decision-focused sensitivity parameters for imperfect data analyses (Wadekar et al., 23 Apr 2025). This suggests that the topic is not a single method but a family of formalisms that replace pointwise sensitivity rankings with bounds, distributions, credible intervals, or risk indicators.

1. Conceptual scope

A recurring formulation is that sensitivity analysis and uncertainty quantification are complementary. One identifies which inputs most strongly affect a response, and the other propagates input uncertainty through the model to quantify output uncertainty; the key idea is that inputs that are both highly sensitive and highly uncertain dominate output variability (Radaideh et al., 2019). In simulation-oriented reviews, sensitivity analysis is placed alongside uncertainty analysis, uncertainty quantification, optimization, and landscape analysis, with analytical objectives such as factor prioritization, factor fixing, variance reduction, and factor mapping (Bade et al., 5 Jun 2026).

The literature distinguishes local and global regimes. Local methods examine behavior near a nominal point or operating condition and include one-at-a-time perturbations, derivative-based indices, and Taylor expansions. Global methods explore the entire feasible input space and are used when uncertain, nonlinear, and interaction-heavy models make local slopes insufficient (Bade et al., 5 Jun 2026). Practical comparisons similarly treat global sensitivity analysis as the relevant framework when the goal is to characterize how output uncertainty may be allocated to sources of uncertainty in model inputs across the entire input space, whereas local analysis gives only point-specific information (Francom et al., 13 Jun 2025).

The objects treated as uncertain vary by domain. In imprecise Markov chains, the uncertain quantities are the initial distribution and transition probabilities (Cooman et al., 2014). In probabilistic supervised learning, the uncertainty enters through the predictive distribution and its Fisher information geometry (Paananen et al., 2019). In causal inference with continuous treatments, the uncertainty is hidden confounding governed by a sensitivity parameter $\Lambda$ (Jesson et al., 2022). In runtime robotics, it is propagated estimator covariance and geometric conditioning inside a Gauss–Newton optimizer (Gaus et al., 16 Dec 2025). In decision analysis with imperfect data, the uncertainty lies in measurement error, missingness, sample selection bias, or record linkage errors, and the primary target becomes the stability of the final decision rather than the numerical estimate alone (Wadekar et al., 23 Apr 2025).

2. Mathematical formulations

One common formulation is linearized uncertainty propagation. For deterministic uncertainty quantification based on sensitivity profiles, the output covariance is written as

$C_y = S\,C_x\,S^T,$

where $C_x$ is the covariance matrix of inputs, $C_y$ is the covariance matrix of outputs, and $S$ is the sensitivity matrix (Radaideh et al., 2019). This formulation is useful, but the cited literature repeatedly emphasizes that it is only one point in a broader design space.

Variance-based global sensitivity analysis is another major formulation. In the cardiovascular study, the total-order Sobol index is given by

$$S_{T_i} = 1 - \frac{\mathrm{Var}\!\left(\mathbb{E}[Y \mid X_{\sim i}]\right)}{\mathrm{Var}(Y)},$$

so that each parameter’s total contribution, including interactions, is measured relative to the total output variance (Thiel et al., 17 Sep 2025). In Bayesian networks, the same first-order and total-effect Sobol decomposition is used to show that one-at-a-time derivatives can miss higher-order effects induced by simultaneous uncertainty in multiple CPT entries (Ballester-Ripoll et al., 2024).

A different line of work makes sensitivity distribution-aware rather than variance-only. For Bayesian supervised learning, the local index

$$\mathrm{R\text{-}sens}(\mathbf{x}^*,x_d,\alpha) \equiv \sqrt{ \alpha \left( \frac{\partial \boldsymbol{\lambda}^*}{\partial x_d^*} \right)^T \mathcal{I}(\boldsymbol{\lambda}^*(\mathbf{x}^*)) \left( \frac{\partial \boldsymbol{\lambda}^*}{\partial x_d^*} \right) }$$

measures how the predictive distribution changes under a perturbation of one predictor, with $\mathcal{I}(\boldsymbol{\lambda}^*)$ the Fisher information matrix of the predictive distribution. The corresponding interaction measure $\mathrm{R\text{-}sens}_2$ uses second derivatives of the predictive parameters (Paananen et al., 2019). This replaces a derivative of the predictive mean by a Fisher-information-weighted norm of derivatives of predictive parameters.

Set-valued probability models lead to lower and upper expectations. In finite Markov chains with imprecise initial and transition probabilities, the basic uncertainty models are credal sets, and time evolution is studied through lower and upper expectations rather than single transition matrices (Cooman et al., 2014). This provides bounds on expectations and probabilities compatible with available information.

Measure-perturbation formulations shift the uncertainty from parameters to probability laws. For random PDEs, the Wasserstein distance

$$d_p(\mu,\nu) := \inf_{\pi\in \Pi(\mu,\nu)} \left( \iint_{X\times X} d_X(x,y)^p\,\pi(dx,dy) \right)^{1/p}$$

is used to quantify perturbations of input measures, and the law of the random solution is studied as a pushforward $C_y = S\,C_x\,S^T,$0 under the solution operator (Ernst et al., 2020). In causal inference, hidden confounding is bounded by the continuous treatment-effect marginal sensitivity model through a density-ratio envelope controlled by $C_y = S\,C_x\,S^T,$1: $C_y = S\,C_x\,S^T,$2 This yields lower and upper bounds on dose-response quantities rather than point identification (Jesson et al., 2022).

3. Methodological families

Practical reviews organize the available techniques into several families. Widely used approaches include local one-at-a-time perturbation, Morris screening, Sobol variance decomposition, regression-based sensitivity analysis, derivative-based global sensitivity measures, Shapley values, Delta indices, and design-of-experiments screening methods (Radaideh et al., 2019, Francom et al., 13 Jun 2025, Bade et al., 5 Jun 2026). The reviews are explicit that different methods answer different questions, require different assumptions, and differ in computational cost and interpretability.

Variance-based methods allocate output variance to main effects and interactions. In the practical comparison, Sobol’ indices are presented as the default when inputs are independent and variance decomposition is the main goal, while total indices provide compact rankings of inputs together with interaction participation (Francom et al., 13 Jun 2025). Screening methods such as Morris trade exact variance attribution for low cost, and large variability in elementary effects is interpreted as nonlinearity and/or interactions (Radaideh et al., 2019).

When the output distribution matters beyond variance, moment-independent measures become relevant. The white paper highlights Borgonovo’s Delta through

$C_y = S\,C_x\,S^T,$3

where the distributional distance compares unconditional and conditional output laws. This is useful when skewness, multimodality, or tail behavior matter and a variance-only summary would be incomplete (Bade et al., 5 Jun 2026). The practical review similarly treats Delta indices as appropriate when one cares about changes in the full output distribution rather than only output variance (Francom et al., 13 Jun 2025).

A further generalization is sensitivity with respect to input distribution parameters rather than inputs themselves. In that framework, uncertain inputs are modeled as $C_y = S\,C_x\,S^T,$4, sensitivities are computed with the likelihood ratio/score function method, and the resulting normalized sensitivity matrix is converted into an eigenvalue problem. The leading eigenvectors identify directions of simultaneous perturbation of distribution parameters that most change the output uncertainty metric (Yang, 2022). The paper explicitly connects this construction to Fisher information and relative entropy, and shows that moment sensitivity, failure probability sensitivity, output-density sensitivity, and information-theoretic sensitivity are special cases of a single utility-based formalism.

4. Dynamic, stochastic, and mechanistic systems

In stochastic process models, uncertainty-aware sensitivity analysis often replaces a single model by a family of compatible models. For finite discrete-time Markov chains, uncertain initial and transition probabilities are represented by credal sets, producing an imprecise Markov chain. The time evolution can be propagated efficiently with lower and upper expectations, and under a regularity condition the long-run behavior converges to a uniquely invariant credal set, yielding a non-trivial generalisation of the classical Perron-Frobenius theorem (Cooman et al., 2014). This makes sensitivity analysis part of the state evolution itself, not an external post-processing step.

Mechanistic kinetics models provide a different example. For point defect kinetics equations under irradiation, perturbation theory is used to derive first-, second-, and third-order corrections to vacancy and interstitial concentrations under uncertainty in rate constants such as recombination and sink-loss rates. The reported analysis can accurately predict the solution up to $C_y = S\,C_x\,S^T,$5 deviation in key rate constants and also supports aggregated uncertainty from multiple rate constants (Jin et al., 2023). The empirical result that defect concentrations are more sensitive to changes in the recombination rate $C_y = S\,C_x\,S^T,$6 than to changes in the vacancy-sink rate $C_y = S\,C_x\,S^T,$7 shows how uncertainty-aware sensitivity analysis can separate dominant mechanisms in nonlinear reaction systems.

For differential equations with random inputs, the uncertainty model can itself be uncertain. The Wasserstein analysis of elliptic diffusion establishes Lipschitz-continuity of the probability measure of the solution random field with respect to perturbations of the input measures, and extends the same stability to Lipschitz continuous quantities of interest and coherent risk functionals (Ernst et al., 2020). This means that one can study not only how a PDE propagates randomness, but also how sensitive that propagated uncertainty is to misspecification or approximation of the input law.

Bayesian networks supply a discrete probabilistic counterpart. Instead of modifying conditional-probability entries one at a time, the global analysis treats selected CPT entries as uncertain simultaneously, encodes them as additional variables, compresses the expanded potentials with low-rank tensor decomposition, and computes Sobol indices on the transformed network. The paper reports that Sobol indices can significantly differ from one-at-a-time indices, thereby revealing the true influence of uncertain parameters and their interactions (Ballester-Ripoll et al., 2024). This is a direct illustration of the broader point that higher-order effects can dominate once uncertainty is treated jointly.

5. Predictive uncertainty, runtime risk, and adaptive control

In Bayesian prediction, uncertainty-aware sensitivity analysis can target the predictive distribution or its uncertainty decomposition. The Rényi-divergence framework differentiates the predictive distribution rather than a point prediction and uses the Fisher information of predictive parameters to define local importance and interaction measures (Paananen et al., 2019). A complementary construction in Bayesian neural networks with latent variables computes the average absolute gradient of the predictive mean, the epistemic standard deviation, and the aleatoric standard deviation with respect to each input feature, thereby ranking inputs by how strongly they affect distinct components of predictive uncertainty (Depeweg et al., 2017). The reported toy example identifies $C_y = S\,C_x\,S^T,$8 as epistemically important and $C_y = S\,C_x\,S^T,$9 as aleatorically important, matching the known data-generating structure.

Information-theoretic analysis gives yet another notion of sensitivity. In well-specified Bayesian learning, predictive epistemic uncertainty under log loss is expressed as $C_x$0, and the paper introduces the test-train sensitivity

$C_x$1

which measures how informative the $C_x$2-th training point is about the test point given the rest of the training data (Futami et al., 2023). This suggests a precise formalization of the common claim that epistemic uncertainty should be small when a test point is similar to the training data.

Runtime estimation systems use sensitivity information as a proxy for imminent failure. SUPER, a framework for visual-inertial odometry and SLAM, propagates uncertainty via sensitivities extracted from the Schur complement blocks of the Gauss-Newton normal matrix, combines propagated uncertainty with residual magnitude and geometric conditioning, and forms a real-time risk indicator that is backend agnostic (Gaus et al., 16 Dec 2025). The reported experiments show prediction of trajectory degradation 50 frames ahead with an improvement of $C_x$3 to the baseline, stop or relocalization triggering with $C_x$4 recall, and less than $C_x$5 additional CPU cost. In this setting, uncertainty-aware sensitivity analysis is not merely diagnostic; it is coupled directly to runtime risk assessment.

Related logic appears in uncertainty-aware learning under nuisance variation and exploration control. In high-energy physics, an uncertainty-aware classifier takes the nuisance parameter $C_x$6 as an input feature and is later profiled over in the likelihood fit, rather than being forced to ignore nuisance dependence (Ghosh et al., 2021). In continuous-control reinforcement learning, PPO-UE introduces uncertainty-aware exploration based on the ratio uncertainty level $C_x$7, and the sensitivity study shows that performance depends strongly on this control parameter, with the best test performance reported at $C_x$8 and the best training reward at $C_x$9 (Zhang et al., 2022).

6. Decision-centered analysis, assumptions, and disputes

Several recent formulations move the target of analysis from outputs to decisions. The confidence-in-decision framework begins with a decision rule, varies assumptions about data imperfections using a sensitivity parameter, and constructs a metric that records whether the decision changes and, if not, how similar the resulting inferences remain (Wadekar et al., 23 Apr 2025). In the election example, the baseline decision remains “challenger wins” for $C_y$0, whereas in the lead-exposure example the intervention decision changes around $C_y$1 under one MNAR mechanism and around $C_y$2 under another. The central point is that an estimate can move while the decision remains stable, or conversely a modest inferential shift can reverse the decision.

Causal inference with continuous treatments similarly replaces point estimates by identified sets indexed by a user-chosen robustness parameter. Under the continuous treatment-effect marginal sensitivity model, the dose-response function $C_y$3, conditional dose-response $C_y$4, and contrasts such as $C_y$5 are bounded above and below by optimizing over admissible latent density-ratio perturbations (Jesson et al., 2022). The paper is explicit that $C_y$6 is not identifiable from data alone; it is a researcher-chosen robustness parameter. This makes uncertainty-aware sensitivity analysis both technically rigorous and substantively dependent on domain judgment.

The methodological literature is equally explicit about assumptions and limitations. Variance-based methods often assume independent inputs, while dependent or constrained inputs require alternative devices such as Shapley values, Delta indices, ALE, or constrained optimal-design approaches (Francom et al., 13 Jun 2025). The Bayesian-network Sobol framework does not allow multiple uncertain child-state probabilities from the same parent configuration because the simplex constraint induces dependence (Ballester-Ripoll et al., 2024). The Rényi-divergence approach requires a predictive distribution with valid Fisher information and is most appropriate for probabilistic models with well-calibrated uncertainty (Paananen et al., 2019). SUPER assumes homoscedastic first-order propagation in the small-residual regime and notes that singular information matrices or multiplicative speckle noise can violate its assumptions (Gaus et al., 16 Dec 2025).

Broader reviews add a governance perspective. Sensitivity auditing is presented as a complementary framework emphasizing transparency, assumption tracking, and responsible model use, with a seven-rule checklist that includes guarding against rhetorical uses of mathematics, adopting an assumption-hunting attitude, identifying sensitive assumptions early, aiming for transparency, and performing state-of-the-art uncertainty and sensitivity analyses (Bade et al., 5 Jun 2026). This suggests that uncertainty-aware sensitivity analysis has both a numerical role and an epistemic role: it quantifies how assumptions propagate, and it exposes which assumptions are carrying the analysis.

Taken together, the literature portrays uncertainty-aware sensitivity analysis as a shift from perturbing fixed models to propagating and interrogating uncertainty in the model specification itself. Across credal Markov chains, Bayesian predictive distributions, Sobol analyses over posterior samples, Wasserstein perturbations of input laws, runtime risk indicators, and decision-centered robustness metrics, the common structure is the same: uncertainty is treated as part of the object of analysis rather than as an external nuisance [(Cooman et al., 2014); (Paananen et al., 2019); (Ernst et al., 2020); (Wadekar et al., 23 Apr 2025); (Thiel et al., 17 Sep 2025)]. A plausible implication is that the main methodological divide is no longer between “sensitivity analysis” and “uncertainty quantification,” but between formulations that collapse uncertainty to a single nominal configuration and formulations that preserve it through the entire inferential or decision pipeline.