Defusing Bertrand’s Paradox

Abstract

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.

Bertrand’s Paradox Resolution and Its Implications for the Bing–Fisher Problem

“…In the time since Bertrand's analysis, it is surprising that the simple answer to Bertrand's problem, which is based on the maximum entropy principle as applied to the mathematical measure of chords, has somehow eluded discovery heretofore. Many scholars instead debated the relative merits of the three solutions provided by Bertrand (e.g., [2,7,15,[22][23][24][25][26][27]). Opinions about Bertrand's three solutions varied among these theorists.…”

“…In essence, these writers agreed with Bertrand without drawing conclusions about the principle of indifference or maximum entropy. Gyenis and Rédei [27] did not challenge any of Bertrand's solutions, but they questioned whether Bertrand's paradox met their standard for a philosophical paradox. As they stated in a philosophical journal, The interpretation proposed here should make clear that Bertrand's Paradox cannot be "resolved" -not because it is an unresolvable, genuine paradox but because there is nothing to be resolved: the "paradox" simply states a provable, non-trivial mathematical fact, a fact which is perfectly in line both with the correct intuition about how probability theory should be used to model phenomena and how probability theory is in fact applied in the sciences.…”

Is there a Bayesian justification of hypothetico‐deductive inference?

“…10 The best known of these is Bertrand's paradox; see Gyenis and Rédei (2014) for a recent treatment.…”

What You See Is What You Get

“…Interestingly in [13] Jaynes proposes a solution to this latter by considering essentially the sort of invariance under automorphisms that we detail here in the discrete case. Naturally this has not met with total acceptance and the debate continues, see for example [14][15][16][17][18][19][20][21]. )…”