Here’s an accuracy-theoretic argument for probabilism (the thesis that only probabilities are rationally admissible credences) on finite spaces that does not make any continuity assumptions on the scoring rule. I will assume all credence functions take values on [0,1].
All probabilities are rationally admissible credences.
If any non-probabilities are rationally admissible, then all non-probabilities satisfying Normalization (whole space has credence 1) and Subadditivity (P(A) + P(B) ≤ P(A∪B) when A and B are disjoint) are rationally admissible with the appropriate prevision being given by a level set integral [correction: actually, I need LSI↑, not the version of LSI in the earlier blog post].
A rationally appropriate scoring rule s satisfies strict propriety for all rationally admissible credences with an appropriate prevision: if V is an appropriate prevision then Vus(u) is better than Vus(v) whenever u and v are different rationally admissible credences.
There is a rationally appropriate scoring rule.
But now we have a cute theorem:
- On any finite space Ω with at least two points, no scoring rule satisfies strict propriety for the credences with Normalization and Subadditivity and level set integral prevision.
It follows no non-probabilities are rationally admissible.
Is this a good argument? I find (2) somewhat plausible—it’s hard to think of a less problematic weakening of the axioms of probability than from Additivity to Subadditivity, and I have not been able to find a better prevision than the level set integral one. Standard arguments for probabilism assume strict propriety for all probabilities. But it seems to me that a non-probabilist will find strict propriety for all probabilities plausible only insofar as they find strict propriety for all admissible credences plausible. Thus (3) is dialectically as good as the usual strict propriety assumption.
I think the non-probabilist’s best way out is to deny strict propriety or to deny that there is a rationally appropriate scoring rule. Both of these ways out work just as well against more standard arguments for probabilism, and I think both are good ways out.
Technically speaking, the advantage of this argument over standard arguments for probabilism is that it makes no assumptions of continuity.
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ReplyDeleteCorrection: While I linked to a post describing the +- version of the level set integral, the "cute theorem" I have proved is actually for the up-arrow version of LSI given here: http://alexanderpruss.com/papers/InconsistentCredences.pdf
ReplyDeleteThe up-arrow version is superior for decision-theoretic purposes, because it commutes with positive affine transformations.
The proof, of course, depends on the fact that "strict propriety for the credences with Normalization and Subadditivity" is a much stronger condition than the classic notion of strict propriety for probabilities. I'll post it when I finish cleaning up the paper in which it sits.
And, finally, here is a written-up version of the proof: http://philsci-archive.pitt.edu/21264/
ReplyDeleteSee Theorem 3 and Corollary 1 in the Appendix.