Bayesian Model Calibration

Bayesian calibration infers uncertain model parameters from data, yielding posterior distributions rather than single best-fit values.

The problem

A simulation model M(x, theta) depends on controllable inputs x and uncertain calibration parameters theta whose true values are unknown. Given experimental observations, calibration asks: what values of theta are consistent with the data, and how confident should we be? Bayesian calibration answers with a full posterior distribution p(theta | data), not a point estimate.

Bayes rule for calibration

Kronos motion — pid vs model

p(theta | data) is proportional to p(data | theta) * p(theta). The prior p(theta) encodes physical knowledge and bounds. The likelihood p(data | theta) measures how well model outputs at theta match observations, given a noise model. The posterior combines both, and its spread quantifies remaining uncertainty in theta after seeing the data.

Components

A forward model or a fast surrogate standing in for it
A prior over calibration parameters
A likelihood, typically Gaussian with observation and model-discrepancy variance
A sampler such as MCMC to explore the posterior

Surrogates make it feasible

MCMC may require tens of thousands of forward evaluations, infeasible for expensive simulations. Replacing M with a trained emulator (Gaussian process or polynomial chaos) makes calibration tractable, but the emulator's own error must be folded into the likelihood variance so the posterior is not falsely narrow.

Identifiability and confounding

Calibration parameters can be unidentifiable when different theta produce indistinguishable outputs, or confounded with model discrepancy (see Model Discrepancy Function). A posterior that stays close to the prior signals the data are uninformative about that parameter. Honest calibration reports these cases rather than presenting an artificially tight estimate. In fusion transport calibration this matters when several closure coefficients trade off against one another, and only combinations, not individual values, are constrained by the data.