Version 1
: Received: 1 November 2024 / Approved: 4 November 2024 / Online: 5 November 2024 (10:53:51 CET)
How to cite:
Bergstra, J. A.; Tucker, J. V. On Defining Expressions for Entropy and Cross-Entropy: The Entropic Transreals and Its Fracterm Calculus. Preprints2024, 2024110144. https://doi.org/10.20944/preprints202411.0144.v1
Bergstra, J. A.; Tucker, J. V. On Defining Expressions for Entropy and Cross-Entropy: The Entropic Transreals and Its Fracterm Calculus. Preprints 2024, 2024110144. https://doi.org/10.20944/preprints202411.0144.v1
Bergstra, J. A.; Tucker, J. V. On Defining Expressions for Entropy and Cross-Entropy: The Entropic Transreals and Its Fracterm Calculus. Preprints2024, 2024110144. https://doi.org/10.20944/preprints202411.0144.v1
APA Style
Bergstra, J. A., & Tucker, J. V. (2024). On Defining Expressions for Entropy and Cross-Entropy: The Entropic Transreals and Its Fracterm Calculus. Preprints. https://doi.org/10.20944/preprints202411.0144.v1
Chicago/Turabian Style
Bergstra, J. A. and John V Tucker. 2024 "On Defining Expressions for Entropy and Cross-Entropy: The Entropic Transreals and Its Fracterm Calculus" Preprints. https://doi.org/10.20944/preprints202411.0144.v1
Abstract
Classic formulae for entropy and cross entropy contain operations 0x and log2 x that are 1 not defined on all inputs. This can lead to calculations with problematic subexpressions such as 2 0 log2 0 and uncertainties in large scale calculations; partiality also introduces complications in logical 3 analysis. Instead of adding conventions, or splitting formulae into cases, we create a new algebra 4 of real numbers with two symbols ±∞, for signed infinite values, and a symbol named ⊥ for the 5 undefined. In this resulting arithmetic, entropy, cross-entropy, Kullback-Leibler divergence, and 6 Shannon divergence can be expressed without any further conventions concerning. The algebra may 7 form a basis for probability theory more generally.
Computer Science and Mathematics, Probability and Statistics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
