On the Interpretability of Conditional Probability Estimates in the Agnostic Setting

TL;DR

This paper investigates the interpretability and calibration of conditional probability estimates in binary classification under the agnostic setting, providing theoretical justification and new measures for their reliability.

Contribution

It introduces a novel calibration measure, establishes uniform convergence results, and offers theoretical insights into the estimation and calibration of conditional probabilities.

Findings

Proposes a new calibration measure for conditional probabilities.

Establishes uniform convergence between empirical and true calibration measures.

Provides theoretical justification for using calibrated probabilities in decision-making.

Abstract

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 Theory. In this paper, we define a novel measure for the calibration property together with its empirical counterpart, and prove an uniform convergence result between them. This new measure enables us to formally justify the calibration property of conditional probability estimations, and provides new insights on the problem of estimating and calibrating conditional probabilities.