  Calibration is a common experimental physics problem, whose goal is to infer
the value and uncertainty of an unobservable quantity Z given a measured
quantity X. Additionally, one would like to quantify the extent to which X and
Z are correlated. In this paper, we present a machine learning framework for
performing frequentist maximum likelihood inference with Gaussian uncertainty
estimation, which also quantifies the mutual information between the
unobservable and measured quantities. This framework uses the Donsker-Varadhan
representation of the Kullback-Leibler divergence -- parametrized with a novel
Gaussian Ansatz -- to enable a simultaneous extraction of the maximum
likelihood values, uncertainties, and mutual information in a single training.
We demonstrate our framework by extracting jet energy corrections and
resolution factors from a simulation of the CMS detector at the Large Hadron
Collider. By leveraging the high-dimensional feature space inside jets, we
improve upon the nominal CMS jet resolution by upwards of 15%.
