Uncertainty Quantification Methods

• Uncertainty Quantification methods are systematic approaches that decompose and assess model uncertainty into epistemic (model) and aleatoric (data) components.
• They employ scalable approximation schemes, Monte Carlo estimators, surrogate models, and deep ensemble techniques to compute predictive uncertainty effectively.
• These methods are crucial for enhancing reliability, safety, and interpretability in machine learning and applied sciences.

Uncertainty quantification (UQ) encompasses the systematic characterization and estimation of uncertainty in predictions and inferences derived from mathematical or computational models. Rigorous UQ methods are central to reliability, safety, and interpretability in probabilistic machine learning and the applied sciences. Contemporary UQ frameworks discriminate between epistemic uncertainty (stemming from limited knowledge or model uncertainty) and aleatoric uncertainty (originating from inherent stochasticity or noise in the data), each requiring distinct methodological treatment. Recent advances integrate scalable approximation schemes, Monte Carlo estimators, surrogate modeling, and high-dimensional inference engines to facilitate practical and theoretically grounded UQ in modern machine learning and scientific computing contexts (Ajirak et al., 7 Sep 2025).

1. Theoretical Foundations: Decomposition of Predictive Uncertainty

The total predictive uncertainty in a probabilistic model is canonically decomposed according to the law of total covariance. Given observed data $\mathcal{D} = \{(x_n, y_n)\}_{n=1}^N$ and a probabilistic model with predictive distribution

$$p(y_* \mid x_*, \mathcal{D}) = \int p(y_* \mid x_*, \theta)\, p(\theta \mid \mathcal{D})\, d\theta,$$

the predictive mean and covariance are

$$\mathbb{E}[y_* \mid x_*, \mathcal{D}] = \mathbb{E}_{p(\theta \mid \mathcal{D})}[\mathbb{E}_{p(y_* \mid x_*, \theta)}[y_*]],$$

$$\operatorname{Cov}[y_* \mid x_*, \mathcal{D}] = \underbrace{\operatorname{Cov}_{p(\theta \mid \mathcal{D})}[\mathbb{E}_{p(y_* \mid x_*, \theta)}[y_*]]}_{\text{Epistemic}} + \underbrace{\mathbb{E}_{p(\theta \mid \mathcal{D})}[\operatorname{Cov}_{p(y_* \mid x_*, \theta)}[y_*]]}_{\text{Aleatoric}}.$$

Thus, total predictive variance $\mathcal{T} = \mathcal{E} + \mathcal{A}$, with the epistemic and aleatoric components isolating model and data sources respectively (Ajirak et al., 7 Sep 2025).

2. Principal UQ Methodologies

Gaussian Process Latent Variable Models and Random Fourier Features

Gaussian Process Latent Variable Models (GPLVMs) situate high-dimensional observations $y_n \in \mathbb{R}^{d_y}$ as noisy mappings from a low-dimensional latent $x_n \in \mathbb{R}^{d_x}$ via independent GPs: $$f_d(\cdot) \sim \mathcal{GP}(0, \kappa_d), \quad y_n \mid x_n, f \sim \mathcal{N}(f(x_n), \Sigma_y).$$ Random Fourier Features (RFF) approximate each kernel $\kappa(x, x')$ by inner-products $\phi(x)^\top \phi(x')$, with $\phi(x) \in \mathbb{R}^J$ constructed by trigonometric transforms of random projections. This reduces GP inference to scalable linear algebra in $\mathbb{R}^J$ (Ajirak et al., 7 Sep 2025).

Monte Carlo Estimation of Epistemic and Aleatoric Uncertainty

Monte Carlo estimators approximate the predictive variance through repeated posterior sampling: $$\hat{\sigma}^2_{\mathrm{epi}, d, *} = \frac{1}{M} \sum_{m=1}^M \left[\mu_{\phi}(X^{(m)})^\top \Sigma_{\theta}^{(m)} \mu_{\phi}(X^{(m)}) + \operatorname{Tr}(C_\phi(X^{(m)})\, \Sigma_\theta^{(m)})\right] + \frac{1}{M} \sum_{m=1}^{M} \operatorname{Tr}\left(\frac{1}{L} \sum_{\ell=1}^{L} [\phi(x_*^{(m, \ell)})^\top \mu_\theta^{(m)}]^2\right).$$

$$\hat{\sigma}^2_{\mathrm{ale}, d, *} = \frac{1}{M} \sum_{m=1}^M \frac{1}{L} \sum_{\ell=1}^L \sigma_{d}^{2,(m)}.$$

This estimator captures the uncertainty from training latents, test latents, and GP weights (Ajirak et al., 7 Sep 2025).

Monte Carlo, Surrogate, and Surrogate-Accelerated Methods

Monte Carlo (MC) remains the baseline for UQ in high dimensions but suffers from slow, $O(N^{-1/2})$ convergence (Zhang, 2020). Surrogate-based multi-fidelity methods (e.g., Lasso Monte Carlo) train a sparse linear surrogate via Lasso regularization and combine it with two-level or multifidelity MC for unbiased, accelerated UQ with error rates $O(\mathrm{Var}[f - f_L]/N)$ (Albà et al., 2022). These hybrid approaches enable UQ in settings where standard MC is computationally infeasible.

3. Application-Specific Instantiations and Best Practices

Deep Learning and Model Uncertainty

Modern deep learning UQ stratifies methods along epistemic/aleatoric axes and sources of uncertainty (He et al., 2023):

• Bayesian Neural Networks (variational or MCMC),
• MC Dropout (variational but scalable),
• Deep Ensembles (approximate posterior via model diversity),
• Heteroscedastic regression (for aleatoric prediction variance).

Quantitative benchmarks show that in some regimes, deep ensembles provide the best combination of predictive accuracy and robust UQ, especially for OOD detection and misclassification recognition (Caldeira et al., 2020, Manivannan et al., 14 Mar 2024). However, sophisticated epistemic UQ (variational, ensemble) may not markedly outperform predictive-entropy from a softmax CNN for all applications, particularly in cross-subject EEG BCI (Manivannan et al., 14 Mar 2024).

UQ in LLMs

Uncertainty in LLMs is now routinely quantified via:

• Direct Model Confidence Prompts (“On a scale from 0 to 1, how confident…”), which have demonstrated best-in-class calibration and efficiency (Zhou et al., 26 Sep 2025, Rivera et al., 13 Jan 2024).
• Sample-based and semantic-consistency approaches (semantic entropy, NLI-based eccentricity), which, despite their sophistication, are often outperformed by direct elicitation in real-world claim-verification (Zhou et al., 26 Sep 2025).
• White-box architectures such as RAUQ, which exploit intrinsic transformer attention-head statistics, providing near-instantaneous and unsupervised UQ for hallucination detection, with lower computational cost and superior rejection curves compared to sampling- and information-theoretic methods (Vazhentsev et al., 26 May 2025).

High-Dimensional Surrogate and Multi-Fidelity UQ

High-dimensional scientific computing (nuclear engineering, CFD, multi-physics) leverages surrogate-based strategies—Lasso, polynomial chaos expansions (PCE), and Gaussian process surrogates (Kriging)—for UQ with tractable costs [(Albà et al., 2022); (Kumar et al., 2022); (Mittal et al., 2014)]. Multifidelity frameworks couple high- and low-fidelity solvers, often via control variate schemes (e.g., MLMC, MFMC), for efficient error control (Zhang, 2020). Hybrid frameworks decompose complex multi-physics or networked systems into modules, each with its local UQ scheme, coupled by a global polynomial chaos representation [(Mittal et al., 2014); (Surana et al., 2011)].

4. UQ Strategies Beyond Standard Probabilistic Models

Information-Theoretic and Robust Optimization Approaches

Information-based UQ methods on Markov Random Fields use the Donsker–Varadhan variational principle to bound QoI prediction biases across KL-divergence-bounded ambiguity sets, leveraging graphical structure to achieve tractable uncertainty certification in high dimensions (Birmpa et al., 2020). The “4th-kind” UQ paradigm synthesizes robust, Bayesian, and decision-theoretic approaches by defining a data-driven likelihood region and minimizing maximal risk via minimum enclosing ball computations in QoI space, thereby balancing tightness of uncertainty intervals and coverage confidence (Bajgiran et al., 2021).

Random Measure and Field-Based UQ

Random measure frameworks (ANOVA decompositions of random counting measures) allow for variance and covariance decomposition in generalized settings, extending to positive random fields and interacting particle systems. Sensitivity indices and structural decompositions precisely localize and quantify uncertainty contributions, complementing algorithmic and model-form UQ (Bastian et al., 2020).

UQ in Hybrid and Dynamical Systems

Hybrid polynomial chaos and transport-theoretic Liouville PDE approaches enable efficient UQ in switching, hybrid, or reset dynamical systems with parametric uncertainty (Sahai et al., 2011). Wavelet-based (Wiener–Haar) expansions, boundary-layer regularization at resets, and characteristic-based PDE solvers capture discontinuities and non-smooth uncertainty propagation infeasible for classical polynomial chaos.

5. Calibration, Evaluation, and Practical Recommendations

Calibration of UQ methods is essential for practical reliability. Empirical scoring (e.g., Brier score, Expected Calibration Error [ECE], prediction interval coverage) must be routinely juxtaposed against theoretical uncertainty decompositions (Rivera et al., 13 Jan 2024). For deep models, direct verbal confidence queries yield surprisingly well-calibrated uncertainty measures when appropriately elicited (Zhou et al., 26 Sep 2025, Rivera et al., 13 Jan 2024). In scientific computing, combining projection-based moments with surrogate or sampling-based propagation provides both efficiency and verifiable uncertainty bounds (Kumar et al., 2022).

Best practices identified include:

• Explicitly separating epistemic and aleatoric contributions in predictive variance estimation.
• Employing surrogate-accelerated or multi-fidelity strategies in high-dimensional or computationally intensive UQ scenarios to avoid the curse of dimensionality (Albà et al., 2022, Zhang, 2020).
• Cautiously interpreting large uncertainty estimates as signals of poor model fit, especially for GPs or surrogates faced with discontinuous or highly non-smooth functions (Ajirak et al., 7 Sep 2025).
• Integrating explicit calibration steps and empirical error analysis, particularly when deploying UQ in safety-critical or real-time settings (Rivera et al., 13 Jan 2024, Vazhentsev et al., 26 May 2025).

6. Open Problems and Directions

Current research targets include systematic integration of variance reduction for Monte Carlo estimators, rigorous non-asymptotic and high-dimensional error guarantees for surrogate-based UQ, scalable hybridizations of model- and data-based uncertainty estimation, and principled approaches for UQ in neural and latent-variable models with complex, multimodal posteriors (Ajirak et al., 7 Sep 2025, Albà et al., 2022). Future extensions to domains such as explainable UQ, hybrid symbolic–probabilistic frameworks, and robust, real-time UQ under distributional and algorithmic drift are deemed critical for the trustworthiness of next-generation machine learning, scientific computing, and engineering design pipelines.