A Maximum Entropy Resolution to the Wine/Water Paradox

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.