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A Maximum Entropy Resolution to the Wine/Water Paradox
Version 1
: Received: 20 June 2023 / Approved: 21 June 2023 / Online: 21 June 2023 (12:38:58 CEST)
A peer-reviewed article of this Preprint also exists.
Parker, M.C.; Jeynes, C. A Maximum Entropy Resolution to the Wine/Water Paradox. Entropy 2023, 25, 1242. Parker, M.C.; Jeynes, C. A Maximum Entropy Resolution to the Wine/Water Paradox. Entropy 2023, 25, 1242.
Abstract
The Principle of Indifference (“PI”: the simplest non-informative prior in Bayesian probability) has been shown to lead to paradoxes since Bertrand (1889). Von Mises (1928) introduced the “Wine/Water Paradox” as a resonant example of a “Bertrand paradox”, and which has been presented as demonstrating that the PI must be rejected. We now resolve these paradoxes by a Maximum Entropy (MaxEnt) treatment of the PI that also includes information provided by Benford’s “Law of Anomalous Numbers” (1938). We show that the PI should be understood to represent a family of informationally-identical MaxEnt solutions; each solution being identified with its own explicitly justified boundary condition. In particular, our solution of the Wine/Water Paradox exploits Benford’s Law to construct a non-uniform distribution representing the universal constraint of scale invariance, which is a physical consequence of the Second Law of Thermodynamics.
Keywords
Scale invariance; Quantitative Geometrical Thermodynamics; Lagrange multipliers
Subject
Physical Sciences, Thermodynamics
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.
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