Kennedy-O'Hagan Framework
The Kennedy-O'Hagan framework calibrates a computer model while explicitly modeling the systematic discrepancy between the model and reality.
The statistical model
Kennedy and O'Hagan (2001) wrote observed reality as z(x) = M(x, theta) + delta(x) + epsilon, where M is the simulator at calibration parameters theta, delta(x) is a discrepancy function capturing systematic model inadequacy, and epsilon is observation noise. Treating delta as a Gaussian process, rather than assuming the model is perfect, is the framework's central contribution.
Why discrepancy matters
If the model has structural error and delta is omitted, the calibration forces theta to absorb that error, producing biased parameter estimates that fit past data but predict poorly. Including delta separates 'wrong parameters' from 'wrong physics,' giving more honest posteriors and wider, better-calibrated predictions.
Joint inference
- Emulator hyperparameters for the surrogate of M
- Calibration parameters theta with physical priors
- Discrepancy GP hyperparameters (length scale, variance)
- Observation noise variance
The identifiability tension
theta and delta are jointly estimated, and the data cannot always separate them: a smooth discrepancy can mimic a parameter shift. Strong, physically grounded priors on delta (for example, requiring it to be small or smooth) are what make the decomposition meaningful. Weak priors leave the split arbitrary.
Use and reporting
The framework is standard in engineering UQ because it produces predictions with defensible uncertainty and flags where a model is structurally inadequate. Kronos design analyses treat any nonzero, structured discrepancy between a reduced model and a high-fidelity run as a signal to investigate the missing physics rather than to retune coefficients. Predictions carry both parameter and discrepancy uncertainty, and the discrepancy field is reported as a diagnostic in its own right.