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A Maximum Entropy Resolution to the Wine/Water Paradox.

Michael C Parker1, Chris Jeynes2

  • 1School of Computer Sciences & Electronic Engineering, University of Essex, Colchester CO4 3SQ, UK.

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|August 26, 2023
PubMed
Summary
This summary is machine-generated.

We resolve Bayesian probability paradoxes by treating the Principle of Indifference (PI) using Maximum Entropy and Benford's Law. This approach reveals the PI as a family of solutions, resolving long-standing issues in statistical inference.

Keywords:
Bayesian probabilityLagrange multipliersquantitative geometrical thermodynamicsscale invariance

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Area of Science:

  • Bayesian probability and statistical inference
  • Thermodynamics and physical constraints

Background:

  • The Principle of Indifference (PI) is a foundational concept in Bayesian probability, often used for non-informative priors.
  • Paradoxes, such as Bertrand's paradox and the Wine/Water paradox, have historically challenged the PI, suggesting its rejection.
  • These paradoxes arise from ambiguities in applying the PI without sufficient justification for boundary conditions.

Purpose of the Study:

  • To resolve the paradoxes associated with the Principle of Indifference in Bayesian probability.
  • To propose a novel framework for understanding and applying the PI using Maximum Entropy and Benford's Law.
  • To demonstrate the physical underpinnings of probability distributions through thermodynamic principles.

Main Methods:

  • A Maximum Entropy (MaxEnt) approach was employed to re-evaluate the Principle of Indifference.
  • Benford's Law of Anomalous Numbers was integrated to provide justified boundary conditions for the PI.
  • The resolution of the Wine/Water Paradox was specifically addressed using this integrated methodology.

Main Results:

  • The Principle of Indifference is shown to represent a family of informationally equivalent Maximum Entropy solutions.
  • Each MaxEnt solution is uniquely identified by an explicitly justified boundary condition.
  • The Wine/Water Paradox is resolved by constructing a non-uniform distribution derived from Benford's Law, reflecting scale invariance.

Conclusions:

  • The paradoxes of the Principle of Indifference are resolved by incorporating Maximum Entropy and Benford's Law.
  • The PI should be understood as a family of MaxEnt distributions, each with a specific, justified boundary condition.
  • Scale invariance, a consequence of the Second Law of Thermodynamics, provides a physical basis for the PI's application in certain contexts.