Propriety and open-mindedness

A scoring rule s(i,r) measures the closeness between one's credence assignment r to a proposition and the truth value, i=0 for false and i=1 for true. I shall assume scoring rules to be continuous. Smaller scores are better.

A scoring rule is proper provided that by your own lights it does not tell you to expect a better (i.e., smaller) score if you just change your credence from r. Given a credence of r in the proposition in question, your expected score from adopting a credence of r' is rs(1,r')+(1−r)s(1,r'). So a proper scoring rule says that this function of r' achieves a minimum at r'=r. A strictly proper scoring rule achieves a minimum only at r'=r.

A scoring rule is open-minded provided that by your own lights it does not tell you to expect worse (i.e., bigger) score if you learn the truth value of some other proposition (we can think of this as the result of a binary experiment). If a scoring rule is not open-minded, then there will be circumstances where score-optimization with respect to one proposition sometimes requires you to shut your ears to other facts. A scoring rule is strictly proper provided that optimizing your score with respect to p requires you to be willing to learn the truth value of any proposition q that by your lights is not independent of p.

As Director of Graduate Studies, I have to attend graduation whenever one of our graduate students gets his Ph.D. On previous occasions, this has been very onerous. But this time I took a notepad and had a lot of fun doing math. In particular, I proved:

• A scoring-rule is proper if and only if it is open-minded.
• A scoring-rule is strictly proper if and only if it is strictly open-minded.

I've since learned from Richard Pettigrew that the implication from propriety to open-mindedness was basically already known. I don't know if the other implication was known as well.

My proof used the standard representation of scoring-rules in terms of convex fucntions, though it turns out that there are simpler proofs at least of the left-to-right implications.

Moreover, the left-to-right implications yield a proof of Good's Theorem. Just use the negative of expected utilities in practical decisions made optimally on the basis of credence r in p, given p and given not p, to define s(1,r) and s(0,r) respectively. It is trivial that this is a proper scoring rule, at least modulo continuity (but the simpler proofs of the left-to-right implications don't use continuity; I think continuity can anyway be proved in this case, but haven't checked details). Hence s is open-minded. But open-mindedness for this rule s is basically what Good's Theorem says.