Embedded Model Form Uncertainty Quantification with Measurement Noise for Bayesian Model Calibration

1. Problem Statement

The paper addresses the critical challenge of calibrating computational models using real-world sensor data when the models are inherently imperfect (inexact).

• Model Inadequacy: All physical models rely on simplifications, leading to systematic discrepancies between model predictions and reality (model form uncertainty).
• The "Overconfidence" Problem: Standard Bayesian calibration often results in posterior distributions that collapse to Dirac-delta functions (point estimates) when models are misspecified. This leads to overconfident predictions that fail to capture the true variability of the system.
• Limitations of Existing Methods:
  • External Correction (Kennedy & O'Hagan): Adds a discrepancy term to the output. While it fits data, the uncertainty cannot be propagated to other Quantities of Interest (QoIs) that depend on the model parameters but not the specific output correction.
  • Internal Correction (Parameter Uncertainty Inflation - PUI): Embeds the inadequacy into the parameters themselves (e.g., Sargsyan's method). This allows uncertainty propagation to any QoI. However, existing PUI methods often struggle with:
    • Measurement Noise: They frequently assume noise-free data or handle noise poorly, leading to biased parameter estimates.
    • Outliers/Offsets: They can be sensitive to data points that cannot be explained by parameter adjustments.
    • Interpretability: Methods like Polynomial Chaos Expansion (PCE) for parameters can be complex and less interpretable.

2. Methodology

The authors propose an Embedded Model Inadequacy framework that treats model bias as a stochastic extension of the latent parameters, combined with new likelihood formulations to handle measurement noise robustly.

A. Embedded Formulation (Internal Correction)

Instead of adding a discrepancy term to the model output, the authors model the latent parameters $\theta$ as random variables:
$$\tilde{\theta} = \theta_m + \delta(\theta_b)$$

• $\theta_m$: The deterministic mean of the parameter.
• $\delta(\theta_b)$: A zero-mean random vector representing the stochastic deviation (model inadequacy), controlled by variance parameters $\theta_b$.
• Distribution: The authors assume $\delta \sim \mathcal{N}(0, \text{diag}(\theta_b))$. This explicit representation is more interpretable than PCE-based approaches and allows for direct propagation of uncertainty to any QoI.
• Forward Model: The stochastic response is approximated using Polynomial Chaos Expansion (PCE) to efficiently compute statistical moments (mean and variance) without expensive sampling at every step.

B. Likelihood Formulations

The paper introduces and compares four likelihood functions designed to match the statistical moments (mean and variance) of the stochastic model predictions with observed data, explicitly accounting for measurement noise ($\sigma_N$):

1. ABC-Likelihood with Noise: An adaptation of Approximate Bayesian Computation (ABC) that matches moments within a tolerance $\epsilon$. The authors derive a specific correction factor $\gamma = \sqrt{\pi/2}$ to correctly relate the expected absolute residual to the standard deviation.
2. Independent Normal (IN) Likelihood: Adapts the classical Gaussian likelihood to include the predicted variance ($\sigma_h$) and measurement noise ($\sigma_N$) in the denominator.
3. Global Moment-matching (GMM) Likelihood (New): Instead of matching moments point-by-point, this treats the set of residuals as samples from a Gaussian Mixture Model. It uses the sampling distributions of the mean (Normal) and variance (Chi-square) of the residuals to construct the likelihood. This reduces sensitivity to individual outliers.
4. Relative Global Moment-matching (RGMM) Likelihood (New): A normalized version of GMM that divides residuals by the predicted standard deviation. This is designed to handle heteroscedasticity (varying noise levels) and prevent high-variance points from dominating the likelihood.

C. Uncertainty Propagation

A key methodological contribution is the full propagation of the joint posterior distribution of the latent parameters ($\theta_m, \theta_b$) to QoIs. Unlike methods that use a single Maximum A Posteriori (MAP) estimator, this approach samples from the full posterior to generate a distribution for the QoI, preserving correlations and the full range of uncertainty.

3. Key Contributions

1. Interpretable Embedding: Replaced complex PCE-based parameter expansions with an explicit additive stochastic model ($\theta_m + \delta$), improving interpretability and separation of fitting tasks.
2. Explicit Noise Handling: Demonstrated that standard ABC likelihoods are highly sensitive to misspecified measurement noise. The paper provides a rigorous derivation for including noise in moment-matching likelihoods and proposes the $\gamma = \sqrt{\pi/2}$ correction.
3. New Likelihoods (GMM & RGMM): Introduced two novel likelihood functions that aggregate information across the dataset rather than fitting point-by-point. These are shown to be more robust to outliers and noise misspecification than standard ABC or IN approaches.
4. Comprehensive Propagation Analysis: Provided a framework for propagating the entire posterior distribution (not just point estimates) to QoIs, enabling a statistical assessment of prediction reliability (e.g., via Z-tests).

4. Results

The methods were evaluated on two cases: a simple 1D linear model and a complex 2D transient thermal simulation of reinforced concrete.

Simple Linear Example

• Noise Sensitivity: ABC was found to be extremely sensitive to noise misspecification. If the prescribed noise was too high or too low, ABC produced biased parameter estimates. GMM, RGMM, and IN were significantly more robust.
• Outliers/Offsets:
  • Offsets (Systematic Bias): All methods adjusted parameters to fit the offset, but RGMM and IN showed different behaviors depending on the variance of the data points.
  • Outliers: ABC was heavily influenced by outliers. RGMM and GMM, by considering the global distribution of residuals, were less skewed by isolated outliers.
• Convergence: ABC required longer MCMC chains to converge due to the strict tolerance $\epsilon$.

Transient Thermal Simulation (Reinforced Concrete)

• Scenario: Calibrating thermal diffusivity in a concrete model that ignores reinforcement bars (model inadequacy) using temperature sensor data. The goal was to predict the cumulative heat flux (a QoI not used in calibration).
• Performance: All likelihoods successfully calibrated the model to cover the temperature data. However, the embedded approach successfully quantified the uncertainty in the heat flux prediction.
• QoI Reliability:
  • Models without embedding produced point estimates for heat flux with zero variance, leading to overconfident (and incorrect) predictions.
  • Models with embedding produced a distribution for the heat flux that successfully enveloped the "true" value generated by the more complex (biphasic) system.
  • Z-Tests: The embedded models yielded Z-values < 1.96 (indicating the true value is within the 95% confidence interval), whereas the non-embedded model failed this test.

5. Significance and Conclusion

This paper provides a robust framework for trustworthy Digital Twins. By embedding model inadequacy directly into the parameters and using noise-aware likelihoods, the authors enable:

• Reliable Uncertainty Quantification: Predictions include both aleatoric (noise) and epistemic (model form) uncertainties.
• Transferability: Uncertainty quantified during calibration can be reliably propagated to any derived Quantity of Interest, not just the calibrated outputs.
• Robustness: The proposed GMM and RGMM likelihoods offer a practical solution to the common problem of noise misspecification and outliers in sensor data, which often plagues existing Bayesian calibration methods.

The work concludes that while the computational cost is higher (due to PCE and full posterior propagation), the resulting predictive distributions are far more informative and reliable for decision-making in engineering applications than traditional point-estimate approaches. Future work should focus on applying these methods to real-world sensor data and exploring correlated inadequacy terms.