Truth directed scoring rules on an infinite space

A credence assignment c on a space Ω of situations is a function from the powerset of Ω to [0, 1], with c(E) representing one’s degree of belief in E ⊆ Ω.

An accuracy scoring rule s assigns to a credence assignment c on a space Ω and situation ω the epistemic utility s(c)(ω) of having credence assignment c when in truth we are in ω. Epistemic utilities are extended real numbers.

The scoring rule is strictly truth directed provided that if credence assignment c₂ is strictly truer than c₁ at ω, then s(c₂)(ω)>s(c₁)(ω). We say that c₂ is strictly truer than c₁ if and only if for every event E that happens at ω, c₂(E)≥c₁(E) and for every event E that does not happen at ω, c₂(E)≤c₁(E), and in at least one case there is strict inequality.

A credence assignment c is extreme provided that c(E) is 0 or 1 for every E.

Proposition. If the probability space Ω is infinite, then there is no strictly truth directed scoring rule defined for all credences, or even for all extreme credences.

In fact, there is not even a scoring rule that strictly truth directed when restricted to extreme credences, where an extreme credence is one that assigns 0 or 1 to every event.

This proposition uses the following result that my colleague Daniel Herden essentially gave me a proof of:

Lemma. If PX is the power set of X, then there is no function f : PX → X such that f(A)≠f(B) whenever A ⊂ B.

Now, we prove the Proposition. Fix ω ∈ Ω. Let s be a strictly truth directed scoring rule defined for all extreme credences. For any subset A of PΩ, define c_A to be the extreme credence function that is correct at ω at all and only the events in A, i.e., c_A(E)=1 if and only if ω ∈ E and E ∈ A or ω ∉ E and E ∉ A, and otherwise c_A(E)=0. Note that c_B is strictly truer than c_A if and only if A ⊂ B. For any subset A of PΩ, let f(A)=s(c_A)(ω).

Then f(A)<f(B) whenever A ⊂ B. Hence f is a strictly monotonic function from PPΩ to the reals. Now, if Ω is infinite, then the reals can be embedded in PΩ (by the axiom of countable choice, Ω contains a countably infinite subset, and hence PΩ has cardinality at least that of the continuum). Hence we have a function like the one the Lemma denies the existence of, a contradiction.

Note: This suggests that if we want strict truth directedness of a scoring rule, the scoring rule had better take values in a set whose cardinality is greater than that of the continuum, e.g., the hyperreals.

Proof of Lemma (essentially due to Daniel Herden): Suppose we have f as in the statement of the Lemma. Let ON be the class of ordinals. Define a function F : ON → A by transfinite induction:

• F(0)=f(⌀)

• F(α)=f({F(β):β < α}) whenever α is a successor or limit ordinal.

I claim that this function is one-to-one.

Let H_α = {F(δ):δ < α}.

Suppose F is one-to-one on β for all β < α. If α is a limit ordinal, then it follows that F is one-to-one on α. Suppose instead that α is a successor of β. I claim that F is one-to-one on α, too. The only possible failure of injectivity on α could be if F(β)=F(γ) for some γ < β. Now, F(β)=f(H_β) and F(γ)=f(H_γ). Note that H_γ ⊂ H_β since F is one-to-one on β. Hence f(H_β)≠f(H_γ) by the assumption of the Lemma. So, F is one-to-one on ON by transfinite induction.

But of course we can’t embed ON in a set (Burali-Forti).