Tuesday, June 25, 2013

Scoring rules and outcomes of betting behavior

The literature has two ways of measuring the fit between one's credence in a proposition and reality. The "epistemic way" uses a scoring rule to measure the distance between one's credence and the truth (if the proposition is true, then a credence of 0.8 is closer to truth than a credence of 0.6). The "pragmatic way" looks at how well one is going to do if one bets in accordance with one's credence.

A standard condition imposed on scoring rules is propriety. A proper scoring rule is one where you don't expect to improve your by shifting your credence without evidence.

I think the two ways come to the same thing, at least in the case of a single proposition. Any appropriate (yeah, some more precision is needed) betting scenario gives rise to a proper scoring rule, where your score for a credence is minus the expected utility on the assumption that you bet according to your credence in the scenario. And, conversely, any proper scoring rule can be generated in this way from an appropriate betting scenario (or at least a limit of them—this is where the details get a bit sketchy).

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