Non-parametric Bayesian Isotonic Calibration: Fighting Over-Confidence in Binary Classification

Abstract

Classifiers can often output a score or a probability indicating how sure they are about the predicted class. Classifier calibration methods can map these into calibrated class probabilities, supporting cost-optimal decision making. Isotonic calibration is the standard non-parametric calibration method for binary classifiers, and it can be shown to yield the most likely monotonic calibration map on the given data, where monotonicity means that instances with higher predicted scores are more likely to be positive. Another non-parametric method is ENIR (ensemble of near-isotonic regression models) which allows for some non-monotonicity, but adds a penalty for it. We first demonstrate that these two methods tend to be over-confident and show that applying label smoothing improves calibration of both methods in more than 90% of studied cases. Unfortunately, label smoothing reduces confidence on the under-confident predictions also, and it does not reduce the raggedness of isotonic calibration. As the main contribution we propose a non-parametric Bayesian isotonic calibration method which has the flexibility of isotonic calibration to fit maps of all monotonic shapes but it adds smoothness and reduces over-confidence without requiring label smoothing. The method introduces a prior over piecewise linear monotonic calibration maps and uses a simple Monte Carlo sampling based approach to approximate the posterior mean calibration map. Our experiments demonstrate that on average the proposed method results in better calibrated probabilities than the state-of-the-art calibration methods, including isotonic calibration and ENIR.