Bayesian Calibration
Update probability distributions over uncertain parameters using data, yielding calibrated parameters that carry their own uncertainty.
Calibration with Uncertainty
Ordinary calibration returns a single best-fit value for each uncertain parameter. Bayesian calibration returns a probability distribution instead. Starting from a prior distribution that encodes what is known before the data, it uses the data to compute a posterior distribution that combines prior knowledge with the evidence. The result is not just a value but a statement of how well the data constrains each parameter.
The Ingredients
- A prior distribution for each parameter, from physics bounds or previous studies.
- A likelihood, describing how probable the observed data is for given parameter values, including measurement error.
- The posterior, proportional to prior times likelihood, computed by sampling methods such as Markov chain Monte Carlo.
What the Posterior Reveals
The posterior width shows how much the data actually pins down a parameter. A narrow posterior means the data is informative; a posterior barely different from the prior means the data says little, and the parameter remains uncertain. Correlations in the posterior reveal parameters the data can only constrain in combination, not individually, a fact that point-estimate calibration hides entirely.
Propagating Forward, and the Firewall
Because the posterior is a distribution, it can be propagated through the model to give predictions that inherit the calibration uncertainty, joining the other terms in the error budget. This is a major advantage over point calibration, which produces confident predictions with no accounting for how uncertain the calibration was. A model discrepancy term can also be included, explicitly representing the gap between model and reality so it is not absorbed into the physical parameters.
The same firewall as ordinary calibration applies: parameters calibrated on a data set must be validated against independent data. Bayesian calibration makes the uncertainty honest, but it does not lift the prohibition on reporting the calibration fit itself as validation.