On Bertrand's paradox

Bangu, Sorin

doi:10.1093/analys/anp124

Cited by 14 publications

(9 citation statements)

References 13 publications

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“…The previous section makes it clear what the possible strategies are to achieve this: One can impose some extra condition on re-labelings that entails either that re-labelings satisfying the extra conditions do not exist (Strategy A) or that the re-labelings satisfying the additional conditions force the re-labelings to be isomorphisms of the probability spaces (Strategy B). Although not formulated in this terminology, Bangu's recent attempt [2] is an example of Strategy A. We show below that Bangu's suggestion for Strategy A is ambiguous however and that resolving the ambiguity makes it either a trivial case of Strategy B or is unsuccessful.…”

Section: Attempts To Save Labeling Invariancementioning

confidence: 82%

“…We also will show that the recent attempt by Bangu [2] to block the emergence of Bertrand's Paradox by requiring re-labelings to preserve randomness cannot succeed non-trivially.…”

Section: The Main Claimsmentioning

confidence: 95%

“…Bertrand's Paradox, published first in [3], is regarded a classical problem in connection with the classical interpretation of probability based on the Principle of Indifference, and it continues to attract interest [17], [24], [2], [22] in spite of alleged resolutions that have been suggested in the large and still growing literature discussing the issue ( [11] and [16] are perhaps the most well-known suggestions for resolutions; the Appendix in [16] contains a brief summary of a number of typical views of the Paradox).…”

Section: The Main Claimsmentioning

confidence: 99%

“…. n} (2) Elementary Principle of Indifference: If the permutation group Πn expresses epistemic indifference about the elementary random events in Xn, then the (Elementary) Classical Interpretation is correct. Thus the (Elementary) Principle of Indifference states that the (elementary version of the) classical interpretation of probability is maintainable only if one is epistemically neutral in some sense about the elementary events.…”

Section: Remarkmentioning

confidence: 99%

“…But if one gives up Distribution Relevance and interprets "randomness" in a way that makes randomness not expressible exclusively in terms of the distributions involved, then the appropriately modified Assumption R constrains even less the emergence of Bertrand's Paradox. Rowbottom and Schackle [22] take Assumption R to be (a technically undefined) "unpredictability" and argue (informally) that there are re-labelings that preserve "unpredictability" and which are not isomorphisms, contrary to what Bangu [2] …”

Section: Attempts To Save Labeling Invariancementioning

confidence: 99%

See 4 more Smart Citations

Defusing Bertrand’s Paradox

Gyenis

Rédei

2015

The British Journal for the Philosophy of Science

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The classical interpretation of probability together with the Principle of Indifference are formulated in terms of probability measure spaces in which the probability is given by the Haar measure. A notion called Labeling Invariance is defined in the category of Haar probability spaces, it is shown that Labeling Invariance is violated and Bertrand's Paradox is interpreted as the very proof of violation of Labeling Invariance. It is shown that Bangu's attempt [2] to block the emergence of Bertrand's Paradox by requiring the re-labeling of random events to preserve randomness cannot succeed nontrivially. A non-trivial strategy to preserve Labeling Invariance is identified and it is argued that, under the interpretation of Bertrand's Paradox suggested in the paper, the paradox does not undermine either the Principle of Indifference or the classical interpretation and is in complete harmony with how mathematical probability theory is used in the sciences to model phenomena; it is shown in particular that violation of Labeling Invariance does not entail that labeling of random events affects the probabilities of random events. It also is argued however that the content of the Principle of Indifference cannot be specified in such a way that it can establish the classical interpretation of probability as descriptively accurate or predictively successful.

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Section: Attempts To Save Labeling Invariancementioning

confidence: 82%

“…We also will show that the recent attempt by Bangu [2] to block the emergence of Bertrand's Paradox by requiring re-labelings to preserve randomness cannot succeed non-trivially.…”

Section: The Main Claimsmentioning

confidence: 95%