The Rationality of Irrationality in the Monty Hall Problem

Enßlin, T. A.; Westerkamp, Margret

doi:10.1002/andp.201800128

Cited by 3 publications

(2 citation statements)

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“…That is, neither a geometric nor an arithmetic addition occurs in either of these cases. As is well-known, the science of probability has always tended to defy intuition by offering unexpected and surprising solutions: the Monty Hall problem (see Enßlin et al 2019 [ 19 ]; Enßlin & Westerkamp 2019 [ 20 ]) is just one example of many; Edwin Jaynes also delighted in exploiting the Maximum Entropy machinery to objectively solve problems (such as the loaded dice; see Jaynes 1978 [ 21 ], Jaynes 1982 [ 22 ]) in a manner aimed at disconcerting those unfamiliar with these methods.…”

Section: Technical Discussionmentioning

confidence: 99%

A Maximum Entropy Resolution to the Wine/Water Paradox

Parker,

Jeynes

2023

Entropy

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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.

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Section: Technical Discussionmentioning

confidence: 99%

A Maximum Entropy Resolution to the Wine/Water Paradox

Parker,

Jeynes

2023

Entropy

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“…Finally, in an article focused on the subtleties of the Monty Hall problem and mostly aiming at information theoretical entertainment, Torsten Enßlin and Margret Westercamp investigate the possibility that human intuition might be more rational than contemporary psychology thinks …”

Section: Informationmentioning

confidence: 99%

The Physics of Information

Enßlin

Jasche

Skilling³

2019

Annalen der Physik

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InformationWhat is information? A short answer could be: Anything that changes our minds and states of thinking! Information is virtual. It can be carried by speech, an image, scratches on a stone, patterns in a photon field, the connections of neurons in our brains or even by the wavefunction of an electron. Information does not depend on the actual means of transmission nor does its identity change when changing the carrier. Information is physical. It needs to be sustained by a physical substrate and rearranging that substrate to contain the wanted information requires to spend energy or work. Without a physical world, information can not be stored, processed, or transmitted. Our world is permeated with different representations of information. Actually Information about the physical laws governing our Universe can be found everywhere and in everything.Could it then be that the real fundamental elements of this world are tiny bits of information? Could we get it from bit? [1] A number of researcher investigate this possibility, and several of their attempts to recover physical laws from the principles governing information are reported about in this special issue on physics of information. The articles by Gregg Jaeger, [2] by Ariel Caticha, [3] and by John Skilling and Kevin Knuth [4] address the reconstruction of quantum mechanics from information theory. Furthermore, Kevin Knuth and James Walsh [5] show how to recover space-time from particles sending each others signals in an initially geometry-free setting.A central element of most works on information is the concept of a probability. Probabilities can be defined via various routes. Most familiar might be their definitions by the frequentists school as relative frequencies of events (in the limit of infinite amounts of such). A more general definition of probabilities is the Bayesians' one, which regards probabilities as quantifiers of statements, expressing how much those should be expected to be true. Although this definition looks subjective on the first glance, it rests on a solid mathematical theorem by Richard Cox [6] that any extension of binary logic to a continuum of single valued truthfulness quantifier that respects a minimum of requirements must be isomorphic to probability theory. And this definition turns out to embrace the frequentists' definition in case that infinite numbers of comparable events are actually available and can be counted. The majority of authors of this special issues seem to favor the Bayesian perspective, which is probably a result of our own, the guest editors', preference and not necessarily a reflection of the prevailing position of contemporary scientists.The different views on probabilities of frequentists and Bayesians are addressed by the article of Allen Caldwell. [7] He builds a bridge between these antagonistic camps by explaining frequentists' constructions in a Bayesian language. In particular, his argumentation why the omnipresent p-value in statistical and experimental literature might often capture a releva...

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A Maximum Entropy Resolution to the Wine/Water Paradox

Parker¹,

Jeynes²

2023

Preprint

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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.

show abstract

The Rationality of Irrationality in the Monty Hall Problem

Cited by 3 publications

References 5 publications

A Maximum Entropy Resolution to the Wine/Water Paradox

A Maximum Entropy Resolution to the Wine/Water Paradox

The Physics of Information

A Maximum Entropy Resolution to the Wine/Water Paradox

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