This prior is impossible to implement for almost all situations, because there is no way to assign uniform probabilities over an infinite number of things—you just end up with 0 everywhere, which isn’t normalized, and hence is not a probability distribution.
If you have enough data, you can get a proper posterior from an improper prior.
That’s true. I guess it could be your “prior” even if it’s not a probability distribution. You don’t just need “enough” data, then—you need data that gives you quickly-decreasing likelihoods as a function of some parameter that numbers finite bunches of hypotheses. Which I don’t think is all that common.
If you have enough data, you can get a proper posterior from an improper prior.
That’s true. I guess it could be your “prior” even if it’s not a probability distribution. You don’t just need “enough” data, then—you need data that gives you quickly-decreasing likelihoods as a function of some parameter that numbers finite bunches of hypotheses. Which I don’t think is all that common.