More dwindling of the prospects for an accuracy-based argument for probabilism in infinite cases

Let P be the set of countably additive probabilities on a countable set Ω. A strictly proper accuracy scoring rule on some set of credences C that includes P is a function s from C to [ − ∞, M]^Ω for some finite M such that E_ps(p)>E_ps(q) for any p ∈ P and any q ∈ C: i.e., from the point of view of each probability in P, that probability has the highest expected accuracy. (It’s a little easier for my example to work with accuracy scoring rules.)

We can identify P with the non-negative functions of ℓ¹(Ω) that sum to 1. This identification induces a topology on P (based on the norm and weak topologies on ℓ¹(Ω) which are the same). On [ − ∞, M]^Ω we will take the product topology.

We say that a scoring rule is uniformly bounded provided there is some finite R such that |s(p)(ω)| < R for all p ∈ C and ω ∈ Ω. We say that one function strictly dominates another provided that the former is strictly less than the latter everywhere.

Theorem: There is a uniformly bounded strictly proper scoring rule s on C that is continuous on P and a credence c ∈ C − P such that s(c) is not strictly dominated by s(p) for any p ∈ P.

In a recent post, I showed that it’s not possible to use strictly proper scoring rules to produce a strict domination argument for probabilism (the thesis that all credences should be probabilities) in infinite cases when we take probabilities to be finitely additive, because in the finitely additive case there is no strictly proper scoring rule. The above Theorem is a complement for the countably additive case: it shows that there are nice strictly proper scoring rules for the countably additive probabilities, but they don’t support the strict domination results that arguments for probabilism seem to require.

There is a remaining open question as to what happens when one further assumes additivity of the scoring rule. But I do not think additivity of a scoring rule is a reasonable constraint, because it seems to me that epistemic utilities will depend on some global features of a forecast.

Here is a sketch of the proof of the theorem. Assume Ω is the natural numbers. Identify P with the non-negative members of ℓ¹ that sum to 1. Let s(p)(n)=p(n)/∥p∥₂ for p ∈ P. Note that the ℓ²-norm is continuous on ℓ¹, and hence s is continuous on Ω. Observe that s(p)∈ℓ¹. By the Cauchy-Schwarz inequality (together with the condition for equality in it), s is strictly proper on P. Define s(c)(n)=1/2(n + 1) for any credence c that is not in P. Note that E_ps(p)=∥p∥₂ for all p. Observe that E_ps(c)<∥s(c)∥₂∥p∥₂ for every p by Cauchy-Schwarz again. But ∥s(c)∥₂<1. Thus, s is strictly proper. But s(c) for c not a probability is not a member of ℓ¹, and hence is not dominated by any score of a probability.