  Accurate calibration of probabilistic predictive models learned is critical
for many practical prediction and decision-making tasks. There are two main
categories of methods for building calibrated classifiers. One approach is to
develop methods for learning probabilistic models that are well-calibrated, ab
initio. The other approach is to use some post-processing methods for
transforming the output of a classifier to be well calibrated, as for example
histogram binning, Platt scaling, and isotonic regression. One advantage of the
post-processing approach is that it can be applied to any existing
probabilistic classification model that was constructed using any
machine-learning method.
  In this paper, we first introduce two measures for evaluating how well a
classifier is calibrated. We prove three theorems showing that using a simple
histogram binning post-processing method, it is possible to make a classifier
be well calibrated while retaining its discrimination capability. Also, by
casting the histogram binning method as a density-based non-parametric binary
classifier, we can extend it using two simple non-parametric density estimation
methods. We demonstrate the performance of the proposed calibration methods on
synthetic and real datasets. Experimental results show that the proposed
methods either outperform or are comparable to existing calibration methods.
