You’re not allowed to do that. With a countably infinite set, your only option for priors that assign everything a number between 0 and 1 is to take a summable infinite series. (Exponential distributions, like that proposed by Peter above, are the most elegant for certain questions, but you can do p(n)=cn^{-2} or something else if you prefer to have slower decay of probabilities.)
In the case with colors rather than integers, a good prior on “first bead color, named in a form acceptable to Omega” would correspond to this: take this sort of distribution, starting with the most salient color names and working out from there, but being sure not to exceed 1 in total.
Of course, this is before Omega asks you anything. You then have to have some prior on Omega’s motivations, with respect to which you can update your initial prior when ve asks “Is it red?” And yes, you’ll be very metauncertain about both these priors… but you’ve got to pick something.
You’re not allowed to do that. With a countably infinite set, your only option for priors that assign everything a number between 0 and 1 is to take a summable infinite series. (Exponential distributions, like that proposed by Peter above, are the most elegant for certain questions, but you can do p(n)=cn^{-2} or something else if you prefer to have slower decay of probabilities.)
In the case with colors rather than integers, a good prior on “first bead color, named in a form acceptable to Omega” would correspond to this: take this sort of distribution, starting with the most salient color names and working out from there, but being sure not to exceed 1 in total.
Of course, this is before Omega asks you anything. You then have to have some prior on Omega’s motivations, with respect to which you can update your initial prior when ve asks “Is it red?” And yes, you’ll be very metauncertain about both these priors… but you’ve got to pick something.
I am happy with that explanation. Thanks.