On the Best Accuracy Arguments for Probabilism

Nielsen, Michael

doi:10.1017/psa.2021.43

Cited by 8 publications

(13 citation statements)

References 4 publications

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“…Thus, p → sup q∈R q, s(p) is the supremum of continuous functions and hence it is lower semicontinuous at these points. (This observation is due to Nielsen [2021]. )…”

Section: Proofsmentioning

confidence: 79%

“…Then Q ∩ G is non-empty. Note that every point z of F satisfies (4) u, z ≤ u, s(u) by propriety, where u(ω) = 1/|Ω| for all ω, and hence every point z of G satisfies (4) as well (a convex combination of points z satisfying (4) will satisfy it as well, and by Lemma 1, so will every point in the closure of Conv F ). Moreover, u, s(u) is finite.…”

Section: Proofsmentioning

confidence: 99%

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Necessary and Sufficient Conditions for Domination Results for Proper Scoring Rules

Pruss¹

2021

Preprint

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Scoring rules measure the deviation between a probabilistic forecast 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. Probabilistic forecasts need not satisfy the axioms of the probability calculus, but Predd, et al. (2009) 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.

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“…Thus, p → sup q∈R q, s(p) is the supremum of continuous functions and hence it is lower semicontinuous at these points. (This observation is due to Nielsen [2021]. )…”

Section: Proofsmentioning

confidence: 79%

Section: Proofsmentioning

confidence: 99%

Necessary and Sufficient Conditions for Domination Results for Proper Scoring Rules

Pruss¹

2021

Preprint

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“…In the finitely additive case, there are no strictly proper scoring rules when the probabilities are defined on the powerset, or when they are defined on the Borel sets of a Polish space. Nielsen [10] has proved an interesting domination result for quasi-strictly proper scoring rules, but requiring quasi-strict propriety stacks the deck against non-probabilistic credences and hence is dialectically inadequate in an argument for probabilism. Moreover, there does not appear to be a good argument for quasi-strict propriety that isn't also an argument for strict propriety.…”

Section: Discussionmentioning

confidence: 99%

“…Recently, Pettigrew [12] announced that this result holds without the assumption of additivity, merely assuming probability-continuity. This proof was shown to have flaws [10], but correct proofs were found by Nielsen [10] and Pruss [15]. Nielsen's proof also extended the result to the 1 A scoring rule is additive if the score at ω ∈ Ω is the sum of scores each of which depends only on the credence for a single event E and whether that event occurs at ω.…”

Section: Introductionmentioning

confidence: 93%

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Accuracy, probabilism and Bayesian update in infinite domains

Pruss

2022

Synthese

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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 probabilismthe doctrine that credences ought to satisfy the axioms of probabilismand 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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