Moment Multicalibration for Uncertainty Estimation

Aaron Roth – University of Pennsylvania


We show how to achieve multi-calibrated estimators not just for means, but also for variances and other higher moments. Informally, this means that we can find regression functions which, given a data point, can make point predictions not just for the expectation of its label, but for higher moments of its label distribution as well — and those predictions match the true distribution quantities when averaged not just over the population as a whole — but also when averaged over an enormous number of finely defined population subgroups. It yields a principled way to estimate the uncertainty of predictions on many different subgroups — and to diagnose potential sources of unfairness in the predictive power of features across subgroups. As an application, we show that our moment estimates can be used to derive marginal prediction intervals that are simultaneously valid as averaged over all of the (sufficiently large) subgroups for which moment multi-calibration has been obtained. We also show how to obtain tight marginal prediction intervals in an online adversarial prediction setting — solving the same problem as conformal prediction, but without any distributional assumptions.