scientiststhesis reblogged your post and added:
Dunno. I think in the mathematical bits I gave…
A few quick notes:
First: I think we’re both talking about the same thing (i.e. when you say “Is that what I did?” the answer is “yes”).
Unless I am missing something here (and I could be), I think there is no difference between the following:
"calculat[ing] the expected utility for all possible distributions and then averag[ing] them according to your prior over these distributions"
"calculat[ing] some ‘weighted distribution’ based on that prior and us[ing] that distribution to calculate your expected utility"
These just correspond to two different orders of doing the integral. The first one corresponds to ∫ (∫x^4 p(x|H) dx) p(H) dH, and the second one corresponds to ∫ x^4 (∫ p(x|H) p(H) dH) dx. These should be equal.
The only reason we might not be able to exchange the order of integration would be if the limits of one of the integrals depended on the integration variable in the other. That’s not true here: the x integral is over all x for any H (all the distributions are distributions over the reals), and the H integral is over all H for any x (for the same reason — no distribution “excludes” any x).
Second: I think we get “swamped” as long as the prior assigns any positive measure to the set of hypotheses with infinite fourth moment. This is easy to see if you do the integral as ∫ (∫x^4 p(x|H) dx) p(H) dH. Sometimes the inner integral is +infinity, and if the region in H-space where this happens has positive measure according to p(H), we get +infinity for the whole integral.
The only reason this might not happen is if there were a corresponding region of H-space where the inner integral worked out to -infinity, so the two would cancel. (In informal, physics-y reasoning that could presumbly be put on a rigorous footing.) That’s why I chose an even moment here: that can’t happen because it’s impossible for a distribution to have a negative fourth moment. There’s an asymmetry here: given this utility function, there are no “infinitely dispreferred” distributions out there to cancel the influence of the “infinitely preferred” distributions with infinite 4th moment.