Inverse Uncertainty Quantification
Inverse UQ infers uncertain model parameters from observed data, returning distributions that reflect what the data can and cannot pin down.
Turning data into parameter knowledge
Forward UQ assumes input uncertainties are known and computes the output. Inverse UQ runs the other way: given measurements of the output, what were the inputs or parameters, and how uncertain are they? The answer is not a single best-fit value but a distribution that captures every parameter combination consistent with the data.
The Bayesian formulation
Bayes' theorem provides the framework: the posterior distribution of parameters is proportional to the likelihood of the data given the parameters times the prior distribution. The prior encodes what was known beforehand, the likelihood measures fit to data, and the posterior is the updated state of knowledge. Inverse UQ is largely the practice of computing and characterizing this posterior.
Computing the posterior
- Markov chain Monte Carlo draws samples from the posterior without needing its normalizing constant
- Sequential Monte Carlo and nested sampling handle multimodal or evidence-computing problems
- Variational inference approximates the posterior with a tractable family for speed
- The ensemble Kalman filter gives a fast Gaussian approximation for high-dimensional problems
Identifiability
Some parameters cannot be determined from the available data - different values fit equally well. Inverse UQ exposes this through wide or correlated posteriors, which is honest and valuable: it tells you the data is silent on that parameter. Ignoring identifiability produces a confident but meaningless point estimate.
The surrogate connection
Posterior sampling can require millions of model evaluations, one per proposed parameter set. This is infeasible with an expensive simulator, so a surrogate emulates the model inside the inference loop. The surrogate's approximation error must be included in the likelihood, or the posterior will be falsely narrow.
In fusion modeling
Inverse UQ calibrates uncertain physics parameters in Kronos models against experimental and literature data, returning posterior ranges rather than tuned single values. This keeps epistemic uncertainty visible and prevents overfitting a model of the machines to sparse observations.