28.1 Bayes is calibrated by construction
Suppose a Bayesian model is given in the form of a prior density p(θ) and sampling density p(y∣θ). Now consider a process that first simulates parameters from the prior, θsim∼p(θ), and then simulates data given the parameters, ysim∼p(y∣θsim). By the definition of conditional densities, the simulated data and parameters constitute an independent draw from the model’s joint distribution, (ysim,θsim)∼p(y,θ). From Bayes’s rule, it follows that for any observed (fixed) data y, p(θ∣y)∝p(y,θ). Therefore, the simulated parameters constitute a draw from the posterior for the simulated data, θsim∼p(θ∣ysim). Now consider an algorithm that produces a sequence of draws from the posterior given this simulated data, θ(1),…,θ(M)∼p(θ∣ysim). Because θsim is also distributed as a draw from the posterior, the rank statistics of θsim with respect to θ(1),…θ(M) should be uniform.
This is one way to define calibration, because it follows that posterior intervals will have appropriate coverage (A. Philip Dawid 1982; Gneiting, Balabdaoui, and Raftery 2007). If the rank of θsim is uniform among the draws θ(1),…,θ(M), then for any 90% interval selected, the probability the true value θsim falls in it will also be 90%. The same goes for any other posterior interval.