Local Uncertainty Quantification (LUQ)

Updated 19 August 2025

Local Uncertainty Quantification (LUQ) is a framework for assessing uncertainty at a granular level, differentiating between aleatoric and epistemic components for precise prediction analysis.
LUQ leverages Bayesian, conformal, and neural techniques to derive pixel-wise, label-wise, and region-specific uncertainty estimates, directly influencing improved model reliability and decision-making.
LUQ enables fine-grained error analysis across diverse domains—such as imaging, finance, and autonomous systems—thereby supporting targeted improvements and risk management.

Local Uncertainty Quantification (LUQ) refers to the rigorous assessment and characterization of uncertainty—statistical, epistemic, model-based, or otherwise—at a local or granular level within a system, model, or dataset. LUQ methods aim to deliver uncertainty estimates that are specific to individual predictions, spatial locations, temporal points, or data samples, thereby enabling informed, fine-grained decision-making, model interpretation, and risk assessment across a broad spectrum of scientific, engineering, and machine learning domains.

1. Core Principles and Definitions

LUQ is grounded in the concept of associating an explicit measure of uncertainty with each localized entity that is analyzed or predicted. This can take the form of:

Pixel-wise credible intervals in image reconstruction, characterizing the uncertainty for each pixel (Cai et al., 2017).
Local credible intervals or sets around individual observations in regression, classification, or generative tasks (Zhang et al., 29 Mar 2024, Kim et al., 16 Aug 2024).
Label-wise decomposition of uncertainty in multiclass classification, enabling uncertainty quantification at the class level (Sale et al., 4 Jun 2024).
Block-wise or subsystem-based uncertainty in quantum many-body systems, e.g., the local quantum uncertainty (LQU), capturing minimal quantum uncertainty in composite systems (Coulamy et al., 2015).

LUQ typically distinguishes between different forms of uncertainty:

Aleatoric uncertainty, reflecting inherent randomness or irreducible noise in the data or system.
Epistemic uncertainty, arising from model misspecification, limited data, or lack of knowledge.

Mathematical formulations across LUQ frameworks often rely on measures such as credible intervals (quantiles), entropy, variance, mutual information, residual analysis, and model ensembles, adjusted to operate “locally” as per the context.

2. Methodological Frameworks in LUQ

2.1 Local Bayesian and Conformal Techniques

LUQ methodologies widely employ Bayesian and conformal inference techniques, adapted for local calibration:

Proximal MCMC for imaging: Introduces non-smooth sparse priors into Bayesian frameworks and obtains local (pixel-wise) credible intervals using quantile statistics from posterior samples. Highest Posterior Density (HPD) regions offer global context but localized error bars for pixels provide spatial detail (Cai et al., 2017).
Bayesian inverse problems with local dimension reduction: Scope the infinite-dimensional parameter to a finite number of random variables using the Karhunen–Loève (KL) expansion, then use MCMC with locally-adaptive screening to produce credible bands for quantities like local volatility surfaces (Yin et al., 2021).
Conformal and split-conformal prediction: Traditional conformal models apply a global calibration. Recent advances partition the predictor space, using regression trees on conformity scores to define regions, then calibrate uncertainty locally within each region to yield tighter, locally-adaptive prediction intervals (Kim et al., 16 Aug 2024). For dynamics models, similar locality-ensuring conformal methods produce region-dependent scaling of predictive covariance, enabling state- and action-specific uncertainty region adaptation (Marques et al., 12 Sep 2024).

2.2 Machine Learning and Neural Estimation

Generative Parameter Sampler (GPS): Applies individual parameterization, modeling each observation with its own parameter generator, yielding scalable, observation-specific UQ suited for LUQ (Shin et al., 2019).
Neural surrogates for corrective estimation: In dynamical systems with local nonlinearities, LUQ is achieved by decomposing the response into nominal (linear) and corrective (neural network–modeled) terms, leading to fast local uncertainty estimation for each realization (De, 2020).
PCS-UQ: Integrates model screening (predictability), bootstrapped assessment of inter-sample and algorithmic variability (stability), and a locally-adaptive multiplicative calibration to yield prediction intervals that respond to local uncertainty, with theoretical and empirical guarantees (Agarwal et al., 13 May 2025).

2.3 Structural Decomposition and Explainability

Label-wise and concept-based decomposition: Decomposes total uncertainty into per-label (class) aleatoric and epistemic components, with variance-based formulations offering interpretability and actionability (Sale et al., 4 Jun 2024). Concept Activation Vector (CAV) approaches associate local uncertainty with interpretable high-level features, enabling explanation and targeted rejection or mitigation (Roberts et al., 5 Mar 2025).

3. Applications of LUQ

LUQ is extensively applied in domains where fine-grained uncertainty is essential:

Radio Astronomy: Pixel-wise credible intervals and HPD regions allow astronomers to distinguish between true celestial features and imaging artefacts in radio interferometric data (Cai et al., 2017).
Computational Finance: Bayesian calibration of local volatility surfaces yields credible bands for price forecasts, with risk management value evidenced by posterior bands encompassing observed market prices (Yin et al., 2021).
Biomedical Imaging: Label-wise uncertainty quantification in tumor classification not only outperforms global entropy metrics in interpretability but informs data acquisition and expert review by identifying high-uncertainty classes or regions (Sale et al., 4 Jun 2024).
Scientific Machine Learning and Dynamical Systems: Neural-network-based LUQ surrogates provide rapid estimation of response statistics under uncertainty for complex, locally nonlinear physical systems, facilitating design and reliability assessment (De, 2020).
Inverse Problems in PDEs: The BiLO framework separates uncertainty over PDE parameters from neural operator solution approximation, enabling efficient local propagation and tractable high-dimensional Bayesian inference (Zhang et al., 22 Jul 2025).
LLMs: LUQ methodologies for long-form text generation use sample diversity and inter-sentence NLI entailment to quantify uncertainty correlating with factuality, aiding in robust response selection, abstention, and ensemble output for improved reliability (Zhang et al., 29 Mar 2024).
Robotics: Local conformal calibration applied to learned or analytical robot dynamics models yields spatially or regionally adaptive uncertainty estimates crucial for planning and safety guarantees under both aleatoric and epistemic uncertainty (Marques et al., 12 Sep 2024).

4. Theoretical Properties and Error Analysis

Many LUQ frameworks are grounded in formal guarantees or properties:

Coverage Guarantees: Add-one-in robustness of adaptive partitions in conformal methods yields finite-sample group-conditional coverage (Kim et al., 16 Aug 2024).
Variance-based Decomposition: Variance-based label-wise measures satisfy axioms such as additivity (A7), invariance under location shifts (A5), and monotonicity under mean-preserving spreads (A3), distinguishing them from entropy-based alternatives (Sale et al., 4 Jun 2024).
Error Propagation in Bilevel Inference: In bilevel operator learning, the dynamic error in posterior sampling and the static error in the posterior distribution are both directly bounded by the lower-level optimization tolerance, establishing a tradeoff between computational accuracy and UQ reliability (Zhang et al., 22 Jul 2025).
Statistical Robustness: Full posterior sampling (via MCMC or bootstraps) integrates parameter, model, and data uncertainty, enabling statistically sound interval estimation even in high-dimensional, non-smooth scenarios (Cai et al., 2017, Agarwal et al., 13 May 2025).

5. Impact, Limitations, and Challenges

The fine granularity achieved by LUQ methods allows practitioners to:

Prioritize data collection or labeling in regions of high local epistemic uncertainty (Sale et al., 4 Jun 2024).
Improve scientific interpretation in big-data and ill-posed inverse problems by providing localized error bounds (Cai et al., 2017).
Enhance safety and reliability of autonomous systems and planning algorithms via locally-calibrated risk controls (Marques et al., 12 Sep 2024).
Support selective prediction and abstention policies in high-risk applications, increasing system trustworthiness (Zhang et al., 29 Mar 2024).
Provide actionable explainability by decomposing uncertainty into interpretable factors or concepts for targeted intervention (Roberts et al., 5 Mar 2025).

Key limitations include the computational cost of high-dimensional local UQ, which motivates dimension reduction (e.g., KL expansion (Yin et al., 2021)), surrogate modeling (De, 2020), and efficient approximations for large models (Agarwal et al., 13 May 2025). Interpretation of local UQ outputs, especially under model misspecification (Shin et al., 2019, Ahn et al., 2023), and the susceptibility of uncertainty measures to adversarial attacks (Ledda et al., 2023) present further challenges.

6. Recent Innovations and Future Directions

Notable recent directions in LUQ research include:

Adaptive and Ensemble Methods: Adaptive partitioning (e.g., robust regression trees) and model ensembling reduce prediction set size without sacrificing coverage, and allow leveraging strengths of distinct models (Kim et al., 16 Aug 2024, Zhang et al., 29 Mar 2024).
Explainable Uncertainty: Concept activation and sensitivity methods bridge quantitative UQ and interpretability, linking high uncertainty regions to human-interpretable concepts and bias mitigation (Roberts et al., 5 Mar 2025).
Local Calibration for Safety in Planning: Spatially local calibration of predictive distributions achieves probabilistically safe planning in uncertain, time-varying, or adversarial settings (Marques et al., 12 Sep 2024, Ledda et al., 2023).
Framework-Agnostic Uncertainty Quantification: PCS-UQ’s veridical data science methodology offers a paradigm combining model screening, bootstrap aggregation, and local adaptive calibration, providing robustness to model misspecification and selection effects (Agarwal et al., 13 May 2025).
Enhancement of Local-to-Global Integration: Learning Uncertain Quantities (LUQ) as data-driven quantities of interest facilitates measure-theoretic inversion and consistent propagation of both aleatoric and epistemic uncertainties in engineering models (Roper et al., 4 Mar 2024).

Ongoing open problems include improving LUQ robustness under adversarial perturbation (Ledda et al., 2023), scaling LUQ to massive high-dimensional problems with minimal loss in locality or interpretability, and integrating LUQ seamlessly with active learning, fairness, and explainability pipelines in production-grade AI systems.

7. Tables: Representative LUQ Method Categories and Key Features

LUQ Method	Localization Target	Principal Quantification Approach
Pixel-wise QI in Imaging	Pixel	Posterior quantiles via proximal MCMC (Cai et al., 2017)
Block-wise LQU in Quantum Sys.	Block/subsystem	Skew info minimization (Coulamy et al., 2015)
Label-wise Decomposition	Class/label	Variance or entropy decomposition (Sale et al., 4 Jun 2024)
Adaptive Conformal Prediction	Region/subgroup	Local quantile calibration after tree partition (Kim et al., 16 Aug 2024)
Neural/Surrogate Correction	Realization	Nominal + NN-based corrective surrogates (De, 2020)
Concept-based Explanation	Concept/activation	CAVs and Sobol-based sensitivity (Roberts et al., 5 Mar 2025)

References

For details on quantum local uncertainty and phase transitions: "Scaling of the local quantum uncertainty at quantum phase transitions" (Coulamy et al., 2015).
Proximal MCMC with pixel-wise uncertainty intervals: "Uncertainty quantification for radio interferometric imaging: I. proximal MCMC methods" (Cai et al., 2017).
Label-wise local uncertainty in classification: "Label-wise Aleatoric and Epistemic Uncertainty Quantification" (Sale et al., 4 Jun 2024).
Adaptive local conformal calibration: "Adaptive Uncertainty Quantification for Generative AI" (Kim et al., 16 Aug 2024).
Neural correction in dynamical systems: "Uncertainty Quantification of Locally Nonlinear Dynamical Systems using Neural Networks" (De, 2020).
Concept-centric local/global UQ: "Conceptualizing Uncertainty" (Roberts et al., 5 Mar 2025).
LUQ for long-form text: "LUQ: Long-text Uncertainty Quantification for LLMs" (Zhang et al., 29 Mar 2024).
Bilevel local operator learning for PDEs: "BiLO: Bilevel Local Operator Learning for PDE Inverse Problems. Part II: Efficient Uncertainty Quantification with Low-Rank Adaptation" (Zhang et al., 22 Jul 2025).
UQ framework via PCS: "PCS-UQ: Uncertainty Quantification via the Predictability-Computability-Stability Framework" (Agarwal et al., 13 May 2025).