Abstract

We formulate the calibration of predicted probabilities as a problem in Bayesian decision theory. Under an additive Gaussian noise model in logit space, we show that the Bayes-optimal calibration function for the squared-error loss is uniquely characterized as the conditional posterior mean E[p* | p̂], and our main theorem expresses it, under a Gaussian prior, as a sigmoid integral over a Gaussian posterior whose parameters are available in closed form. We further show that the MacKay approximation to this optimal estimator, in the limit of a flat, noninformative prior in logit space, takes the same functional form as temperature scaling, and we derive the correspondence between the temperature parameter and the noise variance. We do not claim that the exact estimator and temperature scaling are identical as estimation procedures. We also prove that the Bayes-optimal estimator under the Brier score reduces to the same posterior mean, providing a unified theoretical basis for squared-type losses. Finally, numerical experiments under two conditions — synthetic logit data in which the assumption holds by construction, and data mediated by logistic regression in which departures from the assumption are diagnosed empirically — corroborate the predicted theoretical behavior and quantify the range over which temperature-scaling-type correction is effective. This verification presupposes that the true logit is analytically known; extending noise-level estimation to real data, where it is not, remains an open problem.