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Keywords = entropic transreals

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13 pages, 258 KiB  
Article
On Defining Expressions for Entropy and Cross-Entropy: The Entropic Transreals and Their Fracterm Calculus
by Jan A. Bergstra and John V. Tucker
Entropy 2025, 27(1), 31; https://doi.org/10.3390/e27010031 - 2 Jan 2025
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Abstract
Classic formulae for entropy and cross-entropy contain operations x0 and log2x that are not defined on all inputs. This can lead to calculations with problematic subexpressions such as 0log20 and uncertainties in large scale calculations; partiality also [...] Read more.
Classic formulae for entropy and cross-entropy contain operations x0 and log2x that are not defined on all inputs. This can lead to calculations with problematic subexpressions such as 0log20 and uncertainties in large scale calculations; partiality also introduces complications in logical analysis. Instead of adding conventions or splitting formulae into cases, we create a new algebra of real numbers with two symbols ± for signed infinite values and a symbol named ⊥ for the undefined. In this resulting arithmetic, entropy, cross-entropy, Kullback–Leibler divergence, and Shannon divergence can be expressed without concerning any further conventions. The algebra may form a basis for probability theory more generally. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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