A Maximum Entropy Resolution to the Wine/Water Paradox
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’, which has been presented as demonstrating th...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-08-01
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| Series: | Entropy |
| Subjects: | |
| Online Access: | https://www.mdpi.com/1099-4300/25/8/1242 |
| _version_ | 1797584853936373760 |
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| author | Michael C. Parker Chris Jeynes |
| author_facet | Michael C. Parker Chris Jeynes |
| author_sort | Michael C. Parker |
| collection | DOAJ |
| description | 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’, which has been presented as demonstrating that the PI must be rejected. We now resolve these paradoxes using 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 to 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. |
| first_indexed | 2024-03-10T23:57:35Z |
| format | Article |
| id | doaj.art-b6845b604ab2469f85430816f88ff260 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T23:57:35Z |
| publishDate | 2023-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-b6845b604ab2469f85430816f88ff2602023-11-19T01:00:39ZengMDPI AGEntropy1099-43002023-08-01258124210.3390/e25081242A Maximum Entropy Resolution to the Wine/Water ParadoxMichael C. Parker0Chris Jeynes1School of Computer Sciences & Electronic Engineering, University of Essex, Colchester CO4 3SQ, UKIndependent Researcher, Tredegar NP22 4LP, UKThe 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’, which has been presented as demonstrating that the PI must be rejected. We now resolve these paradoxes using 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 to 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.https://www.mdpi.com/1099-4300/25/8/1242scale invariancequantitative geometrical thermodynamicsLagrange multipliersBayesian probability |
| spellingShingle | Michael C. Parker Chris Jeynes A Maximum Entropy Resolution to the Wine/Water Paradox Entropy scale invariance quantitative geometrical thermodynamics Lagrange multipliers Bayesian probability |
| title | A Maximum Entropy Resolution to the Wine/Water Paradox |
| title_full | A Maximum Entropy Resolution to the Wine/Water Paradox |
| title_fullStr | A Maximum Entropy Resolution to the Wine/Water Paradox |
| title_full_unstemmed | A Maximum Entropy Resolution to the Wine/Water Paradox |
| title_short | A Maximum Entropy Resolution to the Wine/Water Paradox |
| title_sort | maximum entropy resolution to the wine water paradox |
| topic | scale invariance quantitative geometrical thermodynamics Lagrange multipliers Bayesian probability |
| url | https://www.mdpi.com/1099-4300/25/8/1242 |
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