26.1 Bayes is calibrated by construction

Suppose a Bayesian model is given in the form of a prior density \(p(\theta)\) and sampling density \(p(y \mid \theta).\) Now consider a process that first simulates parameters from the prior, \[ \theta^{\textrm{sim}} \sim p(\theta), \] and then simulates data given the parameters, \[ y^{\textrm{sim}} \sim p(y \mid \theta^{\textrm{sim}}). \] By the definition of conditional densities, the simulated data and parameters constitute an independent draw from the model’s joint distribution, \[ (y^{\textrm{sim}}, \theta^{\textrm{sim}}) \sim p(y, \theta). \] From Bayes’s rule, it follows that for any observed (fixed) data \(y\), \[ p(\theta \mid y) \propto p(y, \theta). \] Therefore, the simulated parameters constitute a draw from the posterior for the simulated data, \[ \theta^{\textrm{sim}} \sim p(\theta \mid y^{\textrm{sim}}). \] Now consider an algorithm that produces a sequence of draws from the posterior given this simulated data, \[ \theta^{(1)}, \ldots, \theta^{(M)} \sim p(\theta \mid y^{\textrm{sim}}). \] Because \(\theta^{\textrm{sim}}\) is also distributed as a draw from the posterior, the rank statistics of \(\theta^{\textrm{sim}}\) with respect to \(\theta^{(1)}, \ldots \theta^{(M)}\) 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 \(\theta^{\textrm{sim}}\) is uniform among the draws \(\theta^{(1)}, \ldots, \theta^{(M)},\) then for any 90% interval selected, the probability the true value \(\theta^{\textrm{sim}}\) falls in it will also be 90%. The same goes for any other posterior interval.

References

Dawid, A Philip. 1982. “The Well-Calibrated Bayesian.” Journal of the American Statistical Association 77 (379): 605–10.
Gneiting, Tilmann, Fadoua Balabdaoui, and Adrian E Raftery. 2007. “Probabilistic Forecasts, Calibration and Sharpness.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 69 (2): 243–68.