The paper has just come out online in Synthese.
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.
Comments from a user egregiously failing in the civility required in academic discussion have been deleted and the user has been banned. My responses to these comments have been deleted as well out of fairness to the user. I should, however, note for the sake of anybody who read my comments that in one of my comments I incorrectedly stated that the logarithmic score is not additive, and the user was right to call me out on it, but did so in a manner that was uncivil, and failures of civility are not tolerated.
ReplyDelete(Specifically, for a subset A of Omega, let s_A(c,t)=0 unless A is a singleton and t=1. Then let s_{w}(c,1)=log c({w}). Then the logarithmic score of c is the sum of s_A(c,1_A(w)) as A ranges over the subsets of Omega, and hence is additive in my sense. I was, however, correct that the logarithmic score is not strictly proper when we allow non-probability credences, since it ony depends on the credences at singletons.)