Multivariate Probability Calibration with Isotonic Bernstein Polynomials

Abstract

Multivariate probability calibration is the problem of predicting class membership probabilities from classification scores of multiple classifiers. To achieve better performance, the calibrating function is often required to be coordinate-wise non-decreasing; that is, for every classifier, the higher the score, the higher the probability of the class labeling being positive. To this end, we propose a multivariate regression method based on shape-restricted Bernstein polynomials. This method is universally flexible: it can approximate any continuous calibrating function with any specified error, as the polynomial degree increases to infinite. Moreover, it is universally consistent: the estimated calibrating function converges to any continuous calibrating function, as the training size increases to infinity. Our empirical study shows that the proposed method achieves better calibrating performance than benchmark methods.