More on strong open-mindedness

For the last couple of days I have been exploring what I like to call strongly open-minded accuracy scoring rules. It’s well known that every proper scoring rule is open-minded in the sense that it never requires you to reject free information: the expected epistemic utility of updating on the free information is always at least as good as your current expected epistemic utility. It’s strictly open-minded provided that in non-trivial cases (i.e., when the information has a non-zero probability of having statistical relevance to the credences you are scoring) you are required to accept the free information.

Now there are two reasons why one might accept free information about some proposition q. First, you might be wrong about q: your credence may be high while q is false or your credence might be low while q is true. Second, even if you are right about q, the free information may boost your credence in the right direction. I say that a scoring rule is strongly open-minded provided that it licenses you to accept and update on the free information even if you disregard the first consideration. We can then tack on “strictly” if it requires you to do so in non-trivial cases. In the case of a strongly open-minded scoring rule, your acceptance of free information is not a sign of doubt in your propositions—it is not a way of hedging your bets—and thus is arguably compatible with faith in the propositions being evaluated.

A strongly open-minded scoring rule can also be characterized in the following way. There is a more ordinary kind of epistemic paternalism where I might have reason to block another from receiving free information on the grounds that this information could mislead due to the fact that the other has different likelihoods from the ones I think are right. For instance, if too many people have an unjustified mistrust of Dr. Smith such that they are likely to believe the opposite of what Dr. Smith’s experiments reveal, there is reason to give a grant to someone else, because Dr. Smith’s experiments are likely to lead people away from the truth, for no fault of Dr. Smith’s. Call this likelihood-based paternalism. But there is another kind of motivation of the refusal of free information for another, which we might call pure-risk-based paternalism. Even if someone else has the same likelihoods as you do—trusts Dr. Smith just as you do—perhaps the risk that Dr. Smith’s experiments will, by pure chance, provide evidence away from the truth is enough to justify not funding these experiments.

I’ve been collecting results about these issues. Here’s what I seem t have so far, though I have to emphasize that sometimes the proofs are just in my head and I might be wrong. I will specialize on scoring rules for a single proposition, given as a pair of functions T and F, where T(x) is the value of having credence x when the proposition is true and F(x) is the value of having credence x when the proposition is false.

1. A scoring rule sometimes calls for pure-risk-based paternalism if and only if it is not strongly open-minded.

2. A scoring rule that’s strongly open-minded is open-minded.

3. A scoring rule (T,F) is (strictly) strongly open-minded if and only if xT(x) and (1−x)F(1−x) are both convex.

4. The logarithmic scoring rule is strictly strongly open-minded. The Brier and spherical rules are not strongly open-minded.

5. If a proper scoring rule is generated by the Schervisch-style integral representation T(x) = T(1/2) + ∫_x^1/2(1−t)b(t)dt and F(x) = F(1/2) + ∫_1/2^xtb(t)dt and b is sufficiently differentiable, then the scoring rule is strongly open-minded if and only if the derivative of log b(x) lies between (3x−2)/[x(1−x)] and (3x−1)/[x(1−x)].

6. A strongly open-minded scoring rule whose logarithm is sufficiently differentiable is unbounded.

7. If your credence in a hypothesis H is at least (at most) 1/2, then a proper scoring rule will not call for purely-risk-based epistemic paternalism with respect to someone whose credence is equal to or higher (lower) than yours.

8. If your credence in a hypothesis H is 1/2, then no proper scoring rule calls for purely-risk-based epistemic paternalism for that hypothesis.

9. For any credences p and r such that 1/2 < r and p < r, there is a strictly proper scoring rule and a situation where the scoring rule calls for the individual with credence r to have purely-risk-based epistemic paternalism for that hypothesis.