Uncertainty Quantification and Propagation

Uncertainty propagation is meaningful only after its components are identified and their interpretations kept distinct. Discretization error is not a random draw merely because it is unknown; uncertain physical inputs may be probabilistic; calibration error may be correlated; and model discrepancy can be only conditionally bounded. Combining all four into one standard deviation can create precision that no evidence supports.

Required background. Probability and Inference for Gravity supplies probability models, covariance, sampling, and coverage.

Helpful background. Formulations of General Relativity supplies the continuum and gauge choices; Differential Equations for Relativity supplies numerical-error mechanisms; Computational Relativity supplies the volume-wide workflow; and Mathematical Relativity supplies validity domains that probabilistic summaries do not enlarge.

Decompose before combining

For an observable $Q$, record at least five components: iterative and discretization error, uncertain physical inputs, stochastic or ensemble variability, calibration or measurement uncertainty, and model-form discrepancy. State which are treated as bounded intervals, empirical ensembles, or probability distributions and why.

If $Q(\boldsymbol{\theta})$ is smooth near a reference input and $\boldsymbol{\theta}$ has covariance $\boldsymbol{\Sigma}$, first-order propagation gives

$$\operatorname{Var}(Q) \approx \mathbf J\boldsymbol{\Sigma}\mathbf J^{\mathsf T}, \qquad J_i=\frac{\partial Q}{\partial\theta_i}.$$

Dropping off-diagonal covariance treats the inputs as uncorrelated for this calculation. Independence implies zero covariance when the moments exist, but zero covariance does not imply independence. The distinction matters for nonlinear propagation: if $X$ is a standard normal variable, $X$ and $X^2$ have zero covariance but are plainly dependent. Correlations between masses, spins, calibration gains, or fitted plasma parameters can change the propagated variance substantially. The linear approximation also fails across thresholds, bifurcations, shocks, or multimodal responses; sampling or surrogate methods must then demonstrate their own convergence.

Roy and Oberkampf place verification, validation, and uncertainty within one framework in Roy and Oberkampf 2011, “Uncertainty Framework,” nozzle example, and Conclusions. Their structure supports component separation; it does not make every model error probabilistic.

Results and limitations through 2026-08-18

Component	Established baseline	Strong evidence	Adverse case
Discretization	Resolution sequence and Richardson-type estimate	Added finer run and independent benchmark	Mixed orders or non-asymptotic grids
Parametric input	Correlated sampling from declared distribution	Sample convergence and analytic moments	Tail or dependence misspecification
Stochastic variability	Replicated runs with fixed protocol	Stable quantiles and coverage	Rare-event under-sampling
Calibration	Versioned data model and covariance	Held-out predictive checks	Selection or covariance drift
Model discrepancy	Separate conditional alternatives	External regime test	Discrepancy mimics calibrated parameter

The evidence cutoff is 2026-08-18. The useful controlled position is that probabilistic propagation can quantify declared input and sampling uncertainty inside a demonstrated domain. The competing position is that flexible discrepancy and calibration models are not settled enough to identify physical parameters when both can explain the same residual.

Model discrepancy and calibration

Kennedy and O’Hagan formulate Bayesian calibration with an explicit discrepancy term in Kennedy and O’Hagan 2001, §§2–4. This can protect predictions from pretending the simulator is exact. But when discrepancy has enough flexibility, it can trade against calibration parameters: different parameter–discrepancy pairs explain the same data.

Brynjarsdóttir and O’Hagan show why physically informed discrepancy is consequential in Brynjarsdóttir and O’Hagan 2014, §§2–4. One defensible strategy constrains discrepancy using physics and held-out regimes. Another reports conditional predictions under several model families without assigning a common probability to their spread. Neither strategy licenses an unconditional parameter measurement.

A counterexample should be designed so model discrepancy has the same observable shape as changing one calibrated parameter. If a procedure advertises calibrated coverage for a declared simulation ensemble, narrow intervals that under-cover in that ensemble fail the claim even when posterior computation converges. The ensemble and interval interpretation must be specified before applying this test.

Propagation and coverage checks

Use a manufactured scalar-wave benchmark with one or more correlated uncertain coefficients and a known analytic observable. Compare linear covariance propagation, direct Monte Carlo, and any polynomial-chaos or surrogate result. Increase samples until means, variances, tail quantiles, and effective sample size stabilize. For deterministic quadrature, refine nodes and order.

Coverage asks how often an interval contains the target over a declared repetition of the experiment. A frequentist confidence procedure claims coverage under repeated data at fixed true parameters, with its scope stated. A Bayesian credible interval instead contains a declared fraction of posterior probability for the observed data; that does not generally imply the same coverage at every fixed true parameter.

For exact Bayesian inference with a proper prior, drawing parameters from that prior and then data from the likelihood gives nominal coverage averaged over this joint simulation ensemble. This is the basis of simulation-based calibration, derived in Stan Development Team n.d., User’s Guide, § “Bayes is calibrated by construction”. State whether a coverage study fixes parameters, varies them over an application-specific test set, or draws them from the prior; the three studies answer different questions. Report the coverage fraction with its finite-sample uncertainty. Failure of the advertised coverage is evidence against that claim, but fixed-parameter undercoverage alone does not establish an error in Bayesian posterior computation. Good coverage under the assumed simulator still does not validate that simulator against nature.

For a relativistic production observable, propagate numerical error without double counting. If all input samples share one finite-resolution bias, treating each as an independent discretization draw artificially shrinks it. Correlated extraction, calibration, and surrogate errors likewise remain correlated.

Reproducing the calculation and recognizing failure

The environment record should pin equations, code and dependency revisions, precision, meshes and tolerances, random generator and seeds, input distributions and covariance, calibration release, discrepancy model, sampler, convergence criteria, and scripts. Outputs retain component-wise samples, weights, diagnostics, coverage trials, and conditional results.

A clean rerun would reproduce analytic moments for the manufactured benchmark, demonstrate sample and resolution convergence, test the confounded adverse case, and compare with an independent non-probabilistic bound. This page states the sequence; no absent calculation is claimed.

Reject the finite claim if incompatible uncertainty types are silently added in quadrature, correlations are ignored, sampling tails do not converge, held-out or coverage checks fail, model spread is labeled a calibrated probability without evidence, or discretization bias is resampled as independent noise. The interpretation could change when an independently reproducible benchmark expands the validated domain, a software or hardware change alters numerical behavior, a major negative result appears, or a community comparison changes the baseline.

Check your understanding

Two inputs each have standard deviation one and equal sensitivities $\partial Q/\partial\theta_i=1$. Their correlation is $0.9$. An analysis assumes independence and reports $\sigma_Q=\sqrt{2}$. What does first-order correlated propagation give?

Solution

The variance is $1+1+2(0.9)=3.8$, so $\sigma_Q=\sqrt{3.8}\approx1.95$. The independence assumption understates uncertainty. This calculation remains conditional on local linearity and the covariance model; a threshold or nonlinear response requires a converged sampling check.

References

• Brynjarsdóttir, Jenný, and Anthony O’Hagan. 2014. “Learning about Physical Parameters: The Importance of Model Discrepancy.” Inverse Problems 30, 114007. DOI.
• Kennedy, Marc C., and Anthony O’Hagan. 2001. “Bayesian Calibration of Computer Models.” Journal of the Royal Statistical Society: Series B 63, 425–464. DOI.
• Roy, Christopher J., and William L. Oberkampf. 2011. “A Comprehensive Framework for Verification, Validation, and Uncertainty Quantification in Scientific Computing.” Computer Methods in Applied Mechanics and Engineering 200, 2131–2144. DOI.
• Stan Development Team. “Simulation-Based Calibration Checking.” Stan User’s Guide, version 2.40, n.d.; accessed 20 September 2026. Documentation.