Hierarchical Bayesian Calibration
Hierarchical models calibrate across multiple experiments or units by sharing information through population-level priors.
Pooling information
When several experiments, devices, or operating campaigns each carry their own calibration parameters, a hierarchical model links them: individual parameters theta_i are drawn from a shared population distribution with hyperparameters phi. This partial pooling lets data-rich units inform data-poor ones without forcing all units to share a single value.
Three pooling regimes
- No pooling: each unit calibrated alone, high variance for sparse units
- Complete pooling: one global parameter, ignores real unit-to-unit variation
- Partial pooling: hierarchical, shrinks estimates toward the population mean by an amount the data determine
The structure
The model has three levels: observations given unit parameters, unit parameters given population hyperparameters, and priors on the hyperparameters. Inference (usually MCMC or variational) recovers both individual theta_i and the population phi, along with their uncertainties. Shrinkage is strongest where a unit has little data and weakest where it has much.
Why it helps
Hierarchical calibration produces more stable per-unit estimates, quantifies genuine between-unit variability separately from within-unit noise, and predicts for a new, unseen unit by drawing from the estimated population distribution. This last capability is valuable when planning a device that has not yet been built.
Cautions
The population distribution's form (often Gaussian) is an assumption that should be checked; heavy tails or multimodality break it. Hierarchical models can also be hard to sample, with funnel-shaped posteriors that require reparameterization. As always, report the estimated between-unit variance, since a near-zero value means the hierarchy added nothing and complete pooling would have sufficed.