Pruss, Alexander
(2022)
Accuracy, Probabilism and Bayesian Update in Infinite Domains.
[Preprint]
Abstract
Scoring rules measure the accuracy or epistemic utility of a credence assignment. A significant literature uses plausible conditions
on scoring rules on finite sample spaces to argue for both probabilism---the doctrine that credences ought to satisfy the axioms of
probabilism---and for the optimality of Bayesian update as a response to evidence. I prove a number of formal results regarding scoring
rules on infinite sample spaces that impact the extension of these arguments to infinite sample spaces. A common condition in the
arguments for probabilism and Bayesian update is strict propriety: that according to each probabilistic credence, the expected
accuracy of any other credence is worse. Much of the discussion needs to divide depending on whether we require finite or countable
additivity of our probabilities. I show that in a number of natural infinite finitely additive cases, there simply do not exist
strictly proper scoring rules, and the prospects for arguments for probabilism and Bayesian update are limited. In many natural
infinite countably additive cases, on the other hand, there do exist strictly proper scoring rules that are continuous on the
probabilities, and which support arguments for Bayesian update, but which do not support arguments for probabilism. There may
be more hope for accuracy-based arguments if we drop the assumption that scores are extended-real-valued. I sketch a
framework for scoring rules whose values are nets of extended reals, and show the existence of a strictly proper net-valued scoring
rules in all infinite cases, both for f.a. and c.a. probabilities. These can be used in an argument for Bayesian update, but it is
not at present known what is to be said about probabilism in this case.
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Accuracy, Probabilism and Bayesian Update in Infinite Domains. (deposited 17 Sep 2022 00:44)
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