Local Uncertainty Quantification

• Local uncertainty quantification is a method that provides pointwise error estimates by analyzing uncertainty at individual data elements such as pixels or features.
• It employs techniques like Bayesian marginalization, HPD regions, and adaptive sampling to yield precise local credible intervals and detailed uncertainty measures.
• This approach is critical in applications like scientific imaging, simulation-based inference, and decision-making, where localized risk assessment enhances interpretability and reliability.

Local uncertainty quantification refers to the rigorous characterization of uncertainty in probabilistic inferences or predictions at a fine scale, such as at the level of individual pixels, observations, input features, regions of the feature space, or parameters. This approach stands in contrast to global measures, focusing instead on highly localized regions or aspects of the model, system, or data. Local uncertainty quantification is essential in scientific imaging, simulation-based inference, machine learning, and uncertainty-aware decision making, enabling practitioners to identify, localize, and interpret uncertainty where it matters most.

1. Theoretical Foundations and Motivations

Local uncertainty quantification arises in domains where uncertainty is heterogenous in space, feature space, or over network structures. Motivations stem from the need to provide detailed error bars on physical reconstructions (e.g., image pixels in radio interferometry), to assign credible intervals to local stochastic outputs (e.g., regression or classification outputs at specific input points), or to propagate uncertainty in distributed, component-based systems.

Formally, local uncertainty quantification can refer to:

• Marginalizing the joint posterior or predictive distribution over high-dimensional latent or parameter spaces to obtain credible intervals, quantiles, or other uncertainty measures for localized variables or functions.
• Decomposing uncertainty into local contributions from nodes or cliques in graphical models, or from subdomains in spatial PDEs.
• Explicitly modeling or estimating the variability (variance, entropy) or higher-order moments at specified points or in local neighborhoods, possibly conditional on observed data or input contexts.

Unlike traditional global UQ (e.g., bounding the overall prediction error or model risk), local UQ provides location-by-location, pointwise, or groupwise uncertainty estimates, yielding fine-grained interpretability and supporting robust scientific conclusions.

2. Methodologies for Local Uncertainty Quantification

Local uncertainty quantification leverages a range of algorithmic and mathematical techniques, including:

a) Bayesian Marginalization and Credible Intervals

In high-dimensional Bayesian inverse problems, such as radio interferometric imaging, the full posterior $p(x|y)$ is explored via Markov Chain Monte Carlo (MCMC) sampling. Pixel-wise credible intervals are then computed as sample quantiles for each component: $$[\xi_{i-}, \xi_{i+}]:\quad P(x_i \in [\xi_{i-}, \xi_{i+}] \mid y) = 1 - \alpha$$ with $[\xi_{i-}, \xi_{i+}]$ estimated via order statistics from the posterior samples for each $x_i$ (Cai et al., 2017).

b) Local Credible Regions and HPD Sets

Highest Posterior Density (HPD) credible regions generalize local intervals to global but spatially-defined regions: $C_\alpha = \{x : f(x) + g(x) \leq \gamma_\alpha\}$ where $f$ and $g$ define the prior and likelihood, and $\gamma_\alpha$ is set for the desired coverage. This construction allows for hypothesis testing of image structure: features can be tested by comparing the HPD threshold to the log-posterior value of a surrogate (structure-suppressed) image, providing a principled test for statistical significance of localized features (Cai et al., 2017).

c) Individual Parameterization and Predictive Matching

In hierarchical or generative parameter sampling frameworks (e.g., Generative Parameter Sampler, GPS), each observation $y_i$ is endowed with its own latent parameter $\theta_i$—enabling local uncertainty to be estimated for each instance. A neural generator $G$ maps random noise $Z_i$ to $\theta_i$; the resulting uncertainty at each data point is robust to outliers and scalable to large datasets (Shin et al., 2019).

d) Domain Decomposition and Component-Based Propagation

Uncertainty quantification in complex PDE systems and multiphysics models can be addressed locally—subdomains or components are modeled separately, each propagating its uncertainties under fixed-point relaxation schemes. The network uncertainty quantification (NetUQ) method iteratively solves for local output distributions while communicating only essential dependencies between components (encoded in an adjacency matrix), supporting arbitrary network topologies and scalable parallel implementation (Carlberg et al., 2019).

e) Tree-Structured and Partition-Based Models

Partitioning the feature space into regions with homogeneous uncertainty, as in Uncertainty-Splitting Neural Regression Trees (USNRT), enables targeted local UQ. Levene’s test is used for heterogeneity detection, and region-specific neural networks are trained to model local mean and variance. This explicitly adapts to uncertainty heterogeneity intrinsic to the data, improving calibration and interpretability (Ma et al., 2022).

f) Model-Agnostic Local Explanations

Post-hoc local explanations with quantified uncertainty can be obtained via bootstrapping over neighborhoods of data points, combined with local surrogate models (e.g., polynomials). The resulting uncertainty intervals for feature importances or explanation scores reflect both sampling and local model misspecification uncertainty (Ahn et al., 2023).

3. Techniques and Algorithms in Practical Settings

The implementation of local uncertainty quantification involves several algorithmic innovations:

• Proximal MCMC methods (e.g., MYULA, Px-MALA) handle non-smooth or sparse priors via Moreau–Yosida envelope smoothing and proximal operators, circumventing non-differentiability issues prevalent in high-dimensional inverse imaging (Cai et al., 2017).
• Two-stage adaptive Metropolis algorithms leverage coarse-to-fine likelihood evaluations, accelerating convergence in high-dimensional Bayesian calibrations (e.g., local volatility surface inference) (Yin et al., 2021).
• Recursive and local linearization for NNs allows for efficient estimation of prediction and uncertainty for each query, propagating parameter uncertainty through model linearization and sampling only the low-dimensional output space (Malmström et al., 2023).
• Normalizing flows with conditional parametrization (cKRnet) enable efficient density estimation for local interface parameters in PDE-based domain-decomposed models, sidestepping the curse of dimensionality inherent to joint estimation (Li et al., 4 Nov 2024).
• Local time-stepping and multilevel methods in MC/MLMC frameworks provide computational savings and enable local UQ even in geometrically complex or mesh-refined domains (Grote et al., 2021).
• Physics-inspired functional operators in RKHS offer moment-based local uncertainty estimates that are sensitive to heterogeneity near prediction outputs, surpassing global stochastic moments for error detection under covariate shift (Singh et al., 2021).
• Partitioned conformal prediction adapts calibration intervals locally by data-driven splitting of the input space, achieving local tightening in generative AI tasks while maintaining finite-sample guarantee (Kim et al., 16 Aug 2024).

A summary table of representative methodologies:

Method	Principle	Local UQ Focus
Proximal MCMC [1711...]	Posterior sampling w/ nonsmooth priors	Pixel/feature intervals and HPD
GPS [1905...]	Hierarchical parameterization	Instance-level intervals
NetUQ [1908...]	Component coupling in networks	Subdomain/component uncertainty
USNRT [2212...]	Region partitioning + NN	Region-specific variance
cKRnet [2411...]	Conditional normalizing flows	Interface/conditional densities

4. Quantitative and Statistical Measures

Multiple local UQ metrics and statistical constructs are employed:

• Sample quantiles of posterior draws for marginal credible intervals.
• Local calibration curves and reliability diagrams for prediction probabilities (e.g., in classification).
• ROC–AUC, PR–AUC, ECE, and Brier score for evaluating local predictive uncertainty in classification and regression (Pickering et al., 2022, Singh et al., 2021, Malmström et al., 2023).
• Variance and entropy-based decompositions (aleatoric vs. epistemic uncertainty) computed per-label or per-region; variance-based measures are argued to have superior properties (A0–A7), covering invariance and additivity criteria (Sale et al., 4 Jun 2024).
• Local structure and distribution metrics (e.g., R and NDIP) that assess alignment between predicted uncertainties and errors, providing bounded, interpretable scores for quality assessment (Pickering et al., 2022).

5. Applications, Impact, and Interpretability

Local uncertainty quantification is essential in:

• Radio interferometric imaging for scientifically valid image interpretation: credible intervals, HPD credible regions, and hypothesis tests of image features are now routine (Cai et al., 2017).
• Bayesian option pricing for risk-aware financial decision-making, where uncertainty bands on local volatility inform hedging and risk control (Yin et al., 2021).
• Medical imaging and diagnosis, providing per-class uncertainty that can trigger additional review or data acquisition for critical labels (Sale et al., 4 Jun 2024).
• Component-based engineering simulations and PDEs, where subdomain uncertainty quantification is vital for coupled multiphysics, scalable solvers, and accurate modeling of spatially localized phenomena (Carlberg et al., 2019, Li et al., 4 Nov 2024, Zhang et al., 22 Jul 2025).
• Model explainability, notably feature attribution with uncertainty intervals, for both regulatory compliance and actionable intelligence in real-world deployment (Ahn et al., 2023).
• Calibration and experimental design, by enabling local adaptation of prediction intervals and improved coverage or experiment selection (Kim et al., 16 Aug 2024, Pickering et al., 2022).

Advances in the field yield improved sample and computational efficiency, robustness to outliers and misspecification, and more informative and actionable uncertainty estimates.

6. Limitations and Open Directions

Major limitations and ongoing issues include:

• Scalability in extreme dimensions: Efficient sampling or density estimation (e.g., via conditional flows) is required as the complexity of local parameter spaces increases (Li et al., 4 Nov 2024).
• Model misspecification and surrogate error: Local bootstrapping and variance-based measures attempt to account for these, but further refinement and theory are required for more complex domains (Ahn et al., 2023).
• Hyperparameter sensitivity in neighborhood selection or split criteria for partition-based local UQ.
• Domain adaptation and transferability: How well do local UQ measures generalize to covariate shifts or OOD samples? Functional operator/RKHS-based approaches and selective abstention methods are evolving solutions (Singh et al., 2021, Petersen et al., 13 Feb 2024).
• Balancing computational and statistical accuracy: Approaches such as LoRA, marginalization of only a subset of network parameters, or approximate sampling schemes attempt to bridge this gap in large-scale models (Zhang et al., 22 Jul 2025, Malmström et al., 2023).
• Visual interpretability: Ensuring that local uncertainty maps, intervals, and regions meaningfully inform end-users, with visualization and diagnostic tooling as a key concomitant need (Ma et al., 2022, Cai et al., 2017).

A plausible implication is that as the complexity of inference and decision frameworks grows, local uncertainty quantification will become an increasingly central pillar not only for scientific rigor but also for the interpretability and trustworthiness of AI and simulation-based pipelines.

7. Summary and Outlook

Local uncertainty quantification provides the granularity needed for trustworthy, interpretable, and actionable inference in high-dimensional, heterogeneous, and structured domains. State-of-the-art methodologies integrate advanced MCMC, conditional and partitioned density estimation, model-agnostic bootstrapping, local operator learning, and information-based bounds. The resulting impact spans inverse imaging, scientific computing, finance, medicine, model explainability, and beyond. Ongoing research continues to refine the computational-statistical tradeoffs, generalization properties, and user interpretability of local UQ frameworks, indicating a broad and sustained trajectory for the field across the sciences and AI.