Some possible progress on continuous scoring rules and dominance in an infinite case

On finite sample spaces, we have the Pettigrew-Nielsen-Pruss domination theorem for strictly proper scoring rules that are continuous when restricted to the probabilities that shows that the score of any non-probability is dominated by the score of a probability. Last year, I showed that for a reasonable sense of “continuous”, this is not true on countably infinite sample spaces (when we take probabilities to be countably additive; for if we take probabilities to be finitely additive, there are no strictly proper scoring rules).

In the comments, Ian then suggested that we want our scoring rule to be continuous on all credences, not just the probabilities.

Here are two preliminary responses, though not all the details of the proof of the second have yet been checked, so I could just be wrong.

First, what happens seems to depend on the topology on the space of credences. Credences can be thought of as functions from PΩ to [0,1]. One possibility is to take the space of credences to get the product topology on [0,1]^PΩ. In that case, there is no continuous strictly proper (or even quasi-strictly proper) scoring rule. This follows from the uncountability of PΩ which shows that any countable intersection of neighborhoods of a probability function will contain infinitely many non-probability functions, so that any continuous score will have the property that for every probability there is a non-probability that gets the same score.

But, second, another reasonable topology on [0,1]^PΩ is the ℓ^∞(PΩ) topology. This topology is easily seen to be equivalent on the probabilities to the ℓ¹(Ω) topology (where a probability p on PΩ corresponds to a function p^* ∈ ℓ¹(Ω) defined by p^*(x) = p({x})). The example in my earlier post was a score s that was equal to the spherical score on all the probabilities and s(c)(n) = 1/2(n+1) for any non-probability credence, where we identify Ω with the natural numbers.

Let Q be the space of probability functions on PΩ. Let d(c) = inf_p ∈ Q∥c − p∥_∞ be the distance from c to Q. We can prove that d(c) = 0 iff c is a probability, and d is continuous in our topology. Let ϕ(c) = 0 if d(c) ≥ 1/4 and ϕ(c) = 4d(c) if d(c) < 1/4. This will be a continuous function. Now define s(c)(n) = ϕ(c)/2(n+1) + (1−ϕ(c))c^*(n)/∥c^*∥₂, where c^*(n) = c({n}), and where the second summand is deemed to be zero if ϕ(c) = 1 (regardless of the denominator). I haven’t checked all the details yet, but this s looks continuous to me in the relevant norm, but the domination result is false for any non-probability c. The important point is that the function c ↦ ∥c^*∥₂ is continuous and non-zero for c such that d(c) < 1/4, and that’s one of the points I might yet have an error in.