On the interpretability of conditional probability estimates in the agnostic setting

We study the interpretability of conditional probability estimates for binary classification under the agnostic setting or scenario. Under the agnostic setting, conditional probability estimates do not necessarily reflect the true conditional probabilities. Instead, they have a certain calibration property: among all data points that the classifier has predicted P(Y=1|X)=p, p portion of them actually have label Y=1. For cost-sensitive decision problems, this calibration property provides adequate support for us to use Bayes Decision Rule. In this paper, we define a novel measure for the calibration property together with its empirical counterpart, and prove a uniform convergence result between them. This new measure enables us to formally justify the calibration property of conditional probability estimations. It also provides new insights on the problem of estimating and calibrating conditional probabilities, and allows us to reliably estimate the expected cost of decision rules when applied to an unlabeled dataset.