There’s no principle that says that prior probability of a population exceeding some size N must decrease more quickly than 1/N asymptotically, or any other property of some system. Some priors will have this property, some won’t.
My prior for real-world security lines does have this property, though this cheats a little by being largely founded in real-world experience already. Does my prior for population of hypothetical worlds involving Truman Show style conspiracies (or worse!) have this property? I don’t know—maybe not?
Does it even make sense to have a prior over these? After all a prior still requires some sort of model that you can use to expect things or not, and I have no reasonable models at all for such worlds. A mathematical “universal” prior like Solomonoff is useless since it’s theoretically uncomputable, and also in a more practical sense utterly disconnected from the domain of properties such as “America’s population”.
On the whole though, your point is quite correct that for many priors you can’t “integrate the extreme tails” to get a significant effect. The tails of some priors are just too thin.
There’s no principle that says that prior probability of a population exceeding some size N must decrease more quickly than 1/N asymptotically, or any other property of some system. Some priors will have this property, some won’t.
My prior for real-world security lines does have this property, though this cheats a little by being largely founded in real-world experience already. Does my prior for population of hypothetical worlds involving Truman Show style conspiracies (or worse!) have this property? I don’t know—maybe not?
Does it even make sense to have a prior over these? After all a prior still requires some sort of model that you can use to expect things or not, and I have no reasonable models at all for such worlds. A mathematical “universal” prior like Solomonoff is useless since it’s theoretically uncomputable, and also in a more practical sense utterly disconnected from the domain of properties such as “America’s population”.
On the whole though, your point is quite correct that for many priors you can’t “integrate the extreme tails” to get a significant effect. The tails of some priors are just too thin.