The Fundamental Nature of the Log Loss Function

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The standard loss functions used in the literature on probabilistic prediction are the log loss function, the Brier loss function, and the spherical loss function; however, any computable proper loss function can be used for comparison of prediction algorithms. This note shows that the log loss function is most selective in that any prediction algorithm that is optimal for a given data sequence (in the sense of the algorithmic theory of randomness) under the log loss function will be optimal under any computable proper mixable loss function; on the other hand, there is a data sequence and a prediction algorithm that is optimal for that sequence under either of the two other loss functions but not under the log loss function.
Original languageEnglish
Title of host publicationLecture Notes in Computer Science
EditorsLev Beklemishev, Andreas Blass, Nachum Dershowitz, Berndt Finkbeiner, Wolfram Schulte
Place of PublicationCham
Number of pages12
ISBN (Electronic)978-3-319-23534-9
ISBN (Print)978-3-319-23533-2
Publication statusE-pub ahead of print - 5 Sept 2015

Publication series

NameLecture Notes in Computer Science
PublisherSpringer International Publishing
ISSN (Print)0302-9743


  • algorithmic theory of randomness, mixability, predictive complexity, predictive randomness, probabilistic prediction, proper loss functions

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