As promised, here is my formal example with a "neutral" prior.
Space has four slots: a,b,c,d
Time has five slots: 1,2,3,4,5
Each space/time combination can hold one of these:
D(ead rock), M(onkey), H(uman), P(osthuman)
Each "universe" describes what is at each space/time slot.
Here are three universes: *,#,@
A state description says which universe I'm in, and which space-time slot I occupy. For example, I hope I'm @c4, and I'm glad I'm not #d5.
The prior is equal over all 60 states so defined.
My current information is "I'm in an H at time 4." Some I'm one of {#c4,#d4,@c4,@d4}. Given that, my posterior gives equal probability to universes #,@. These don't change if my information were "I'm an H."
If my info were "I'm an H or P," my posterior would say p(@) = 7/10, and p(#) = 3/10. If my info were "I'm an M, H, or P," I'd say p(*) = 8/24, p(#) = 6/24, p(@) = 10/24.
Just knowing I'm alive suggests (If I were smart enough to realize the implications) that I'm in a universe filled with life. Just knowing that I'm alive and smart suggests that I'm in a universe where intelligence prospers.
If I've understood Nick's preferred DA prior correctly, it gives the universes *,#,@ equal probability conditional on being in a slot filled with a H or P. Except he's not sure if that should instead be just H, or even M,H, or P.
If it's H or P, the priors must satisfy p(#) = (7/3)*p(@). Such a prior assumes "doom" is more probable, and seems designed to exactly counter act my joy that being smart suggests that I'm in a universe filled with smarts. But I'm not at all clear on why I should fiddle with my prior to do this.
"choose a full prior based on other considerations, and
then fiddle with it so being smart won't suggest your
universe has other smarts"?
Robin Hanson
hanson@econ.berkeley.edu http://hanson.berkeley.edu/
RWJF Health Policy Scholar, Sch. of Public Health 510-643-1884
140 Warren Hall, UC Berkeley, CA 94720-7360 FAX: 510-643-2627