Scoring rules measure the deviation between a forecast, which assigns degrees of confidence to various events, and reality. Strictly proper scoring rules have the property that for any forecast, the mathematical expectation of the score of a forecast p by the lights of p is strictly better than the mathematical expectation of any other forecast q by the lights of p. Forecasts need not satisfy the axioms of the probability calculus, but Predd et al. [9] have shown that given a (...) finite sample space and any strictly proper additive and continuous scoring rule, the score for any forecast that does not satisfy the axioms of probability is strictly dominated by the score for some probabilistically consistent forecast. Recently, this result has been extended to non-additive continuous scoring rules. In this paper, a condition weaker than continuity is given that suffices for the result, and the condition is proved to be optimal. (shrink)

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. (shrink)

When a belief has been influenced, in part or whole, by factors that, by the believer's own lights, do not bear on the truth of the believed proposition, we can say that the belief has been, in a sense, arbitrarily formed. Can such beliefs ever be rational? It might seem obvious that they can't. After all, belief, supposedly, “aims at the truth.” But many epistemologists have come to think that certain kinds of arbitrary beliefs can, indeed, be rational. In this (...) paper, I want to do two things. First, I want to show that the claim that arbitrary beliefs can be rational is inconsistent with the conjunction of two other attractive claims: one saying that rationality requires a certain kind of epistemic immodesty, and one saying that rationality forbids certain kinds of self-ascriptions of epistemic luck. Second, I want to evaluate different ways one might respond to this inconsistent triad. I won't defend any response in particular, but I'll draw out some notable costs and benefits of each response, which may help shed light on the question of whether arbitrary beliefs can be rational. (shrink)

In a recent paper, Pettigrew reports a generalization of the celebrated accuracy-dominance theorem due to Predd et al., but Pettigrew’s proof is incorrect. I will explain the mistakes and provide a correct proof.

The best accuracy arguments for probabilism apply only to credence functions with finite domains, that is, credence functions that assign credence to at most finitely many propositions. This is a significant limitation. It reveals that the support for the accuracy-first program in epistemology is a lot weaker than it seems at first glance, and it means that accuracy arguments cannot yet accomplish everything that their competitors, the pragmatic (Dutch book) arguments, can. In this paper, I investigate the extent to which (...) this limitation can be overcome. Building on the best arguments in finite domains, I present two accuracy arguments for probabilism that are perfectly general—they apply to credence functions with arbitrary domains. I then discuss how the arguments’ premises can be challenged. We will see that it is particularly difficult to characterize admissible accuracy measures in infinite domains. (shrink)