Analytical Bounds between Entropy and Error Probability in Binary Classifications

TL;DR

This paper derives exact analytical bounds between entropy and error probability in binary classification, improving upon previous approximations and confirming Fano's bound as exact for all classifiers.

Contribution

It provides closed-form, approximation-free bounds for binary classification errors, confirming Fano's bound as exact and tightening the upper bounds with respect to prior probabilities.

Findings

Fano's lower bound is exact for all classifiers.

Derived tighter upper bounds based on minimum prior probability.

Bounds are expressed in closed-form without approximations.

Abstract

The existing upper and lower bounds between entropy and error probability are mostly derived from the inequality of the entropy relations, which could introduce approximations into the analysis. We derive analytical bounds based on the closed-form solutions of conditional entropy without involving any approximation. Two basic types of classification errors are investigated in the context of binary classification problems, namely, Bayesian and non-Bayesian errors. We theoretically confirm that Fano's lower bound is an exact lower bound for any types of classifier in a relation diagram of "error probability vs. conditional entropy". The analytical upper bounds are achieved with respect to the minimum prior probability, which are tighter than Kovalevskij's upper bound.