Thoughts on Probability

Matt Briggs is discusing Subjective Versus Objective Bayes Versus Frequentism.But he clearly has a preference and I am reluctant to accept without question the labeling and characterization of a position by one who does not hold it. Apparently, the objectivist (at least as represented by Briggs) does not claim the existence of a unique consistent assignment of probabilities to all propositions. Perhaps the freedom claimed by a subjectivist does not really include assigning arbitrary values to the probability of a specific event about which we have a body of relevant information to consider. It may only be to set some assumed underlying (prior?) probabilities within a range of possibilities that can be shown not to significantly affect the conditional probabilities that are deduced from an extended sequence of observations. (Nothing to do with statsig here, just the idea of a limit). Similarly the frequentist may not be constrained to imagine the repetition of a specific event over time, but rather the (Gibbs?) ensemble of possible scenarios consistent with some agreed on collection of past observations (cf Briggs' "official" premises). Are these really all that different?

At the same time, John Carlos Baez has posted on Probability Theory and the Undefinability of Truth
A much more limited goal than that of assigning objective probabilities to all propositions is that of doing so just for the subset of propositions that make arithmetic statements.

The possibility of making a self-consistent assignment of probabilities to arithmetic propositions has apparently been recently established (by Christiano, Yudkowsky, Herreshoff and Mihaly in a preprint at it also appears that there may in fact be many such assignments.

How would such non-uniqueness get reconciled with the idea of objective Bayesian probability?

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