The project aims to develop quality measures for approximations in machine learning and statistics, using tools of probability and mathematical analysis, such as Stein's method and functional inequalities. Approximate inference techniques have been used in the recent years as a way to speed up the learning process, which is particularly important in the era of big data. It is, however, necessary for researchers to be able to measure the error the associated approximations generate. Indeed, wrong uncertainty or point estimates in applications related, for instance, to engineering or epidemics, may have highly negative outcomes.