The problem seems to be that I am placing equal probabilities on each possible integer while you favor numbers closer to 0.
You are not doing so, since it is impossible. No such probability distribution exists. In fact you recognize this by saying there’s a cap somewhere out there, you just don’t know where. Well, this cap means that small numbers (smaller than the cap) have much, much higher probability (i.e. nonzero) than large numbers (those higher than the cap have zero probability).
Maybe this will serve as an intuition pump: suppose you’ve narrowed down your cap to just a few numbers. In fact, just N and 2N. You’ve given them each equal weight. Well, now p(1) = (1/N + 1/2N)/2 = 3⁄4 N, but p(N+k) = 1⁄4 N, and p(2N+k) = 0. The probability goes down as numbers get larger. Determine your priors over all the caps, compute the resulting distribution, and you’ll find p(n) eventually start to decrease.
You are not doing so, since it is impossible. No such probability distribution exists. In fact you recognize this by saying there’s a cap somewhere out there, you just don’t know where. Well, this cap means that small numbers (smaller than the cap) have much, much higher probability (i.e. nonzero) than large numbers (those higher than the cap have zero probability).
Maybe this will serve as an intuition pump: suppose you’ve narrowed down your cap to just a few numbers. In fact, just N and 2N. You’ve given them each equal weight. Well, now p(1) = (1/N + 1/2N)/2 = 3⁄4 N, but p(N+k) = 1⁄4 N, and p(2N+k) = 0. The probability goes down as numbers get larger. Determine your priors over all the caps, compute the resulting distribution, and you’ll find p(n) eventually start to decrease.