Bayesian Inverse Uncertainty Quantification

• Bayesian Inverse Uncertainty Quantification is a probabilistic framework that infers input uncertainties by combining experimental data with prior knowledge.
• It contrasts with forward UQ by updating model parameters based on outputs rather than propagating input uncertainties through the model.
• It employs methods like hierarchical modeling, surrogate assistance, and gradient-based MCMC to address ill-posed inverse problems effectively.

Bayesian Inverse Uncertainty Quantification (IUQ) is the process to inversely quantify input uncertainties based on experimental data, or, in a broader inverse-problem setting, to update a probabilistic description of unknown parameters or states using observations so that the result is a posterior distribution on admissible values rather than only a deterministic best-fit reconstruction (Wu et al., 2021, Wu et al., 2018). In problems written schematically as $A(x)=y$, Bayesian IUQ treats the unknown $x$ and the data model probabilistically, combines prior knowledge with a likelihood, and conditions on observed data to obtain uncertainty-aware inverse solutions; across the recent literature, this includes calibration of physical model parameters, PDE-constrained inversion, hierarchical population models, surrogate-assisted Bayesian computation, and sampling-free conditional-expectation updates (Riis et al., 2023, Matthies et al., 2015).

1. Scope and relation to forward uncertainty quantification

Bayesian IUQ is commonly contrasted with forward UQ. Forward UQ propagates assumed input uncertainty through a forward model to obtain output uncertainty, whereas inverse UQ uses outputs or experimental data to infer the uncertainty in model inputs or parameters (Wu et al., 2021). In the nuclear Best Estimate plus Uncertainty (BEPU) context, this distinction is operational: forward propagation requires input uncertainty models first, yet those inputs are often specified by expert opinion or user self-assessment, so inverse UQ is used to infer or refine them from benchmark or plant data (Wu et al., 2018, Wang, 2024).

This distinction appears in several application areas. In nuclear thermal-hydraulics, the unknowns are often physical model parameters embedded in closure correlations or constitutive relations; in PDE inversion they may be coefficient fields, source terms, or initial conditions; in electrochemical inverse modeling they may be the diffusion coefficient and the transference number; and in inverse optimization they are objective parameters inferred from observed decisions (Alghamdi et al., 2023, Sethurajan et al., 2018, Chan et al., 24 May 2026). The defining point is not the physical domain but the direction of information flow: outputs and observations constrain the inputs.

Bayesian IUQ is also explicitly motivated by ill-posedness. The survey literature stresses that inverse UQ is often ill-posed because many parameter combinations can explain the same data, and the Bayesian reformulation addresses this by treating the unknowns as random variables and returning posterior distributions that encode uncertainty widths and parameter dependence rather than only single optimal values (Wu et al., 2021). In the conditional-expectation literature, this is described as the well-posed Bayesian reformulation of the ill-posed inverse problem (Matthies et al., 2015).

2. Core probabilistic formulations

A standard starting point is the generic inverse problem

$A(x)=y,$

where $x$ is the unknown solution parameter, $y$ is noisy data, and $A$ is the forward operator or model. In Bayesian form, one writes

$x \sim p(x), \qquad y \sim p(y\mid x),$

and, given observed data, obtains the posterior through Bayes’ rule: $$p(x\mid y)=\frac{p(y\mid x)p(x)}{p(y)}, \qquad p(x\mid y=y_{\mathrm{obs}})\propto L(x\mid y=y_{\mathrm{obs}})\,p(x).$$ In CUQIpy this is the core Bayesian inverse uncertainty quantification step: prior knowledge and a measurement model are combined with the observed data to produce a posterior distribution that quantifies uncertainty in the recovered solution (Riis et al., 2023).

A closely related formulation in calibration problems is the model-updating equation

$y_E(x)=y_M(x,\theta^*)+\delta(x)+\epsilon,$

where $y_E$ is the experimental observation, $x$0 is the computer-model prediction, $x$1 is the unknown target parameter vector, $x$2 is model discrepancy, and $x$3 is measurement error (Wu et al., 2018, Wang, 2024). Under Gaussian assumptions, this yields a likelihood-based posterior kernel of the form

$x$4

with total covariance assembled from experimental, discrepancy, and code or emulator uncertainty (Wu et al., 2018).

The literature also contains a formulation that does not take densities as primary. In the conditional-expectation approach, the Bayesian update is posed as an orthogonal projection in $x$5: if $x$6, then for a scalar quantity $x$7,

$x$8

and analogously for vector-valued random variables (Matthies et al., 2015). This leads to filter-type updates such as

$x$9

for the linear Bayesian update, and higher-order polynomial corrections for nonlinear Bayesian updates (Matthies et al., 2015, Litvinenko et al., 2013).

Across these formulations, the central interpretive point is stable: the posterior is not treated as a single optimal parameter value, but as a probability distribution over admissible values. Posterior summaries then include the posterior mean, maximum a posteriori estimate (MAP), credible intervals, posterior variance or standard deviation, and parameter correlations (Riis et al., 2023, Radaideh et al., 2022).

3. Model discrepancy, covariance structure, and hierarchical formulations

A major theme in Bayesian IUQ is that simulation–experiment mismatch cannot generally be attributed to parameter uncertainty alone. The discrepancy term $A(x)=y,$0 is introduced to represent systematic differences between reality and the computer model due to incomplete or inaccurate physics, numerical approximation errors, and other unmodeled effects (Wu et al., 2018). The associated likelihood covariance is often written as

$A(x)=y,$1

or, equivalently in some papers, $A(x)=y,$2 (Wu et al., 2021, Wang, 2024).

The reason this matters is explicit in the calibration literature: if one assumes $A(x)=y,$3, then the model is implicitly treated as reality, and neglecting discrepancy can force the calibration to compensate for model errors by distorting parameter estimates, leading to over-fitting (Wu et al., 2018). In the TRACE/BFBT application, posteriors obtained without discrepancy become much narrower, but this concentration is interpreted as over-fitting rather than credible learning; with discrepancy included, the posterior spreads more realistically because some mismatch is attributed to model form error rather than parameter shifts (Wu et al., 2018). A related PSBT study reports that discrepancy-aware modular Bayesian calibration improves validation RMSE, whereas omitting discrepancy yields narrower posteriors but slightly worse generalization relative to the nominal model (Wang, 2024).

This concern becomes sharper for time-dependent outputs. In transient thermal-hydraulics, the observation covariance is generally non-diagonal because time points are strongly correlated, and the literature emphasizes that treating them as independent can distort the posterior and encourage fitting the wrong part of the trajectory (Wang, 2024). One transient TRACE study, however, does not model discrepancy because only one transient experiment is used, and explicitly identifies this as a limitation of the study rather than a general principle (Xie et al., 2023).

Another development is hierarchical Bayesian IUQ. Instead of assuming one global calibration vector for all experiments, hierarchical models assign experiment-specific parameters drawn from a population distribution governed by hyperparameters (Wang et al., 2023, Wang, 2024). In nuclear thermal-hydraulics this is motivated by the observation that physical model parameters may vary under different experimental conditions or flow regimes. The proposed benefit is partial pooling: local parameters can vary across experiments, but shrinkage toward a common population distribution reduces over-fitting and makes the inference less sensitive to outliers (Wang et al., 2023). This suggests a shift from “single global calibration” toward population-level uncertainty models when databases cover broad operating conditions.

4. Inference algorithms and posterior computation

Bayesian IUQ is methodologically diverse. Exact or approximate posterior computation may use direct Gaussian sampling, Laplace approximation around the MAP, Metropolis–Hastings, preconditioned Crank–Nicolson, ULA, MALA, NUTS, Randomize-then-Optimize / LinearRTO, Gibbs sampling, CWMH, and conjugate or approximate conjugate samplers (Riis et al., 2023). CUQIpy makes this diversity explicit by constructing a BayesianProblem, automatically forming the posterior from prior and likelihood definitions, and selecting structured samplers based on model structure; for hierarchical Gaussian–Gamma models it can, for example, use conditional conjugacy such as x: LinearRTO and s: Conjugate, while for edge-preserving priors such as LMRF or CMRF it can select UGLA or NUTS (Riis et al., 2023).

Gradient-based methods recur when posteriors are strongly correlated or high-dimensional. In a nonlinear gravity anomaly inversion example, NUTS is reported to mix much better than Metropolis–Hastings, illustrating the importance of gradient-based inference for correlated posteriors (Riis et al., 2023). In hierarchical thermal-hydraulics IUQ, NUTS is used because standard random-walk samplers become inefficient when the number of latent group-specific parameters is large (Wang et al., 2023, Wang, 2024). The software-oriented literature also notes that MCMC diagnostics such as effective sample size and $A(x)=y,$4 are available through ArviZ integration (Riis et al., 2023).

Not all Bayesian IUQ relies on MCMC. The conditional-expectation framework develops sampling-free Bayesian updates by approximating the conditional expectation in finite-dimensional polynomial spaces, often using polynomial chaos expansions (PCE), and derives linear Bayesian updates, quadratic Bayesian updates, and filter constructions without Monte Carlo sampling (Matthies et al., 2015, Litvinenko et al., 2013). In this line of work, the update is formulated as a variational projection problem and then discretized in functional or spectral bases.

Recent approximations further broaden the algorithmic repertoire. For inverse PDE problems with reduced-order deep-learning surrogates, one study computes a surrogate-based MAP estimate in latent space and then generates approximate posterior samples by randomizing the objective and solving many perturbed minimization problems, a randomize-then-minimize strategy that avoids repeated expensive PDE solves and repeated likelihood evaluations (Wang et al., 2024). For stochastic inverse problems governed by SPDEs, another study introduces a two-stage optimization method: a stabilized weighted MAP estimate followed by variational inference over a Gaussian approximate posterior to obtain uncertainty widths, together with an error estimate relating reconstruction error to the number of observations (Song et al., 13 Mar 2025). A separate surrogate-assisted framework couples Bayesian optimization for adaptive Gaussian-process construction with Bayesian inversion on the surrogate, using posterior concentration or multimodality as the primary uncertainty-quantification output (Chiappetta et al., 4 Feb 2026).

5. Surrogates, reduced representations, and software ecosystems

Because posterior sampling may require many model evaluations, surrogate modeling is pervasive in Bayesian IUQ. Gaussian-process metamodels are central in modular Bayesian calibration, where one GP may emulate the expensive code and another GP may emulate model discrepancy (Wu et al., 2018, Wu et al., 2018). In the TRACE/BFBT application, the GP surrogate for the code is validated using $A(x)=y,$5 predictivity coefficients and LOOCV errors; the reported computational reduction is from about 23.7 core-days to 36.7 core-minutes when replacing direct TRACE evaluations during MCMC (Wu et al., 2018). PCE surrogates are also used, as in the Spallation Neutron Source target-station study, where a polynomial-chaos surrogate with reported MAE $A(x)=y,$6, RMSE $A(x)=y,$7, and $A(x)=y,$8 supports MCMC-based inversion of mercury equation-of-state parameters (Radaideh et al., 2022).

Reduced-order representations are equally important when the unknown or the output is functional. In PDE-based inverse problems, Karhunen–Loève expansions, step expansions, Matérn KL expansions, and level-set parameterizations are used to regularize the inverse problem and reduce posterior dimension (Alghamdi et al., 2023). In transient thermal-hydraulics, PCA and functional PCA are used to reduce time-dependent quantities of interest before surrogate construction and MCMC (Xie et al., 2023). The FEBA study reports that conventional PCA needed 4 PCs to explain about 95% of variance and 10 PCs for about 99%, whereas functional PCA used 2 PCs for warped data plus 4 PCs for warping functions to exceed 99% variance explained; it also reports that functional PCA reconstructed quenching behavior more accurately than conventional PCA (Xie et al., 2023). Bayesian neural networks are then used to supply surrogate uncertainty as part of $A(x)=y,$9 (Xie et al., 2023).

The software literature formalizes these workflows. CUQIpy represents Bayesian inverse problems through distributions for unknowns and data, automatic posterior construction, geometry-aware samples, and structured sampler selection; its PDE extension adds PDE abstractions, PDEModel, geometry classes, and FEniCS integration, allowing the forward map in the likelihood to be written as $x$0 or $x$1 (Riis et al., 2023, Alghamdi et al., 2023). The package is explicitly designed so that users can write models closely aligned with the mathematical specification of the inverse problem and then obtain posterior means, credible intervals, variance maps, and trace plots (Riis et al., 2023).

Surrogates are not limited to GP, PCE, or neural operators. In a data-driven CVD reactor study, an XGBoost regressor maps mixed-type reactor setup variables to coating-thickness measurements, and weighted Approximate Bayesian Computation with summary statistics is used for likelihood-free posterior inference because the likelihood for the tree-based predictor is intractable (Loachamín et al., 15 Dec 2025). This indicates that Bayesian IUQ can be coupled to nonparametric black-box regressors, provided posterior computation is adapted accordingly.

6. Applications, validation, and recurring methodological issues

Bayesian IUQ has been applied across a wide range of domains. In nuclear thermal-hydraulics it is used for TRACE physical-model parameters in BFBT, PSBT, and FEBA benchmarks, often under BEPU requirements and often with explicit discrepancy modeling or hierarchical pooling (Wu et al., 2018, Wang, 2024, Xie et al., 2023, Wang, 2024). In structural mechanics it has been used to infer mercury equation-of-state parameters for the Spallation Neutron Source first target station from strain data (Radaideh et al., 2022). In electrochemical systems it quantifies uncertainty in constitutive relations inferred from MRI concentration profiles by reconstructing posterior distributions for $x$2 and $x$3, including concentration-dependent forms (Sethurajan et al., 2018). In PDE inversion it appears in Poisson, heat, electrical impedance tomography, photo-acoustic tomography, nonlinear diffusion, and groundwater-flow-type problems (Alghamdi et al., 2023, Wang et al., 2024). More recent work extends the same logic to nuclear data adjustment, stochastic inverse problems for SPDEs, data-driven inverse optimization, and industrial CVD process control (Brady et al., 9 Sep 2025, Song et al., 13 Mar 2025, Chan et al., 24 May 2026, Loachamín et al., 15 Dec 2025).

Validation and prediction are treated as nontrivial extensions of the inverse step. One BEPU-oriented framework integrates inverse UQ with quantitative validation by Bayesian hypothesis testing, computes a Bayes factor from validation data, and converts it into weights for Bayesian model averaging between prior-based and posterior-based predictions (Xie et al., 2021). This reflects a recurring concern in the literature: posterior concentration by itself is not equivalent to predictive credibility, especially if model discrepancy has been ignored or extrapolated outside the calibration domain.

Several methodological issues recur. First, extrapolating a discrepancy model learned in the inverse-UQ domain to validation or prediction domains is explicitly identified as risky, motivating improved modular Bayesian strategies and sequential test source allocation that keep inverse-UQ cases inside the region covered by validation data (Wu et al., 2018, Wu et al., 2018). Second, low prior correlation between experiments and applications does not necessarily imply that an experiment is uninformative: the nonlinear nuclear-data-adjustment benchmark reports that Chadwick, despite low or even negative correlation with some applications, can still provide substantial information gain (Brady et al., 9 Sep 2025). Third, posterior shape can diagnose identifiability: sharply peaked posteriors indicate low epistemic uncertainty, whereas broad or multimodal posteriors indicate nonuniqueness or strong nonlinearity (Chiappetta et al., 4 Feb 2026).

The practical limitations are equally consistent. MCMC can be computationally expensive; surrogate accuracy matters; model discrepancy is hard to learn because there is no direct data for it; transient outputs require careful covariance treatment; and highly nonlinear or strongly correlated problems may require advanced samplers such as NUTS or specialized approximations (Wu et al., 2021, Riis et al., 2023). In function-valued inverse problems, priors and parameterizations must preserve enough regularity for the forward PDE to remain well posed (Sethurajan et al., 2018). In SPDE inversion, covariance matrices induced by the stochastic forward model can become ill-conditioned or singular, motivating stabilized weighting and variational approximations (Song et al., 13 Mar 2025).

Taken together, these studies present Bayesian IUQ not as a single algorithm but as a probabilistic framework whose defining output is a posterior distribution over unknown inputs, parameters, or fields. Its characteristic strengths are the simultaneous treatment of prior information, measurement uncertainty, model discrepancy, and surrogate or code uncertainty; its characteristic difficulties are identifiability, over-fitting, covariance modeling, and computational cost. The literature therefore combines rigorous Bayesian updating with increasingly specialized computational tools, from GP and PCE surrogates to hierarchical priors, automatic sampler selection, likelihood-free inference, and reduced-order latent representations (Riis et al., 2023, Wu et al., 2021).