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Deconstructing Cross-Entropy for Probabilistic Binary Classifiers

AuDIaS-Audio, Data Intelligence and Speech, Escuela Politecnica Superior, Universidad Autonoma de Madrid, Calle Francisco Tomas y Valiente 11, 28049 Madrid, Spain
Author to whom correspondence should be addressed.
Entropy 2018, 20(3), 208;
Received: 22 February 2018 / Revised: 16 March 2018 / Accepted: 18 March 2018 / Published: 20 March 2018
(This article belongs to the Special Issue Entropy-based Data Mining)
In this work, we analyze the cross-entropy function, widely used in classifiers both as a performance measure and as an optimization objective. We contextualize cross-entropy in the light of Bayesian decision theory, the formal probabilistic framework for making decisions, and we thoroughly analyze its motivation, meaning and interpretation from an information-theoretical point of view. In this sense, this article presents several contributions: First, we explicitly analyze the contribution to cross-entropy of (i) prior knowledge; and (ii) the value of the features in the form of a likelihood ratio. Second, we introduce a decomposition of cross-entropy into two components: discrimination and calibration. This decomposition enables the measurement of different performance aspects of a classifier in a more precise way; and justifies previously reported strategies to obtain reliable probabilities by means of the calibration of the output of a discriminating classifier. Third, we give different information-theoretical interpretations of cross-entropy, which can be useful in different application scenarios, and which are related to the concept of reference probabilities. Fourth, we present an analysis tool, the Empirical Cross-Entropy (ECE) plot, a compact representation of cross-entropy and its aforementioned decomposition. We show the power of ECE plots, as compared to other classical performance representations, in two diverse experimental examples: a speaker verification system, and a forensic case where some glass findings are present. View Full-Text
Keywords: Bayesian; cross-entropy; probabilistic; classifier; discrimination; calibration; ECE plot Bayesian; cross-entropy; probabilistic; classifier; discrimination; calibration; ECE plot
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MDPI and ACS Style

Ramos, D.; Franco-Pedroso, J.; Lozano-Diez, A.; Gonzalez-Rodriguez, J. Deconstructing Cross-Entropy for Probabilistic Binary Classifiers. Entropy 2018, 20, 208.

AMA Style

Ramos D, Franco-Pedroso J, Lozano-Diez A, Gonzalez-Rodriguez J. Deconstructing Cross-Entropy for Probabilistic Binary Classifiers. Entropy. 2018; 20(3):208.

Chicago/Turabian Style

Ramos, Daniel, Javier Franco-Pedroso, Alicia Lozano-Diez, and Joaquin Gonzalez-Rodriguez. 2018. "Deconstructing Cross-Entropy for Probabilistic Binary Classifiers" Entropy 20, no. 3: 208.

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