Given multiverse hypothesis (universes with different physical constants / laws), the number of universes with infinitely large set of laws is much larger (both being infinite, though) than number of universes with finite sets of laws.
The question is not which set is larger, which is in any case almost meaningless since both are infinite, but which set has larger probability measure.
I do think it’s meaningful to talk about different sizes of infinity (for example, countable vs. uncountable), but probability measure is probably more relevant.
To expand on that point—what you are refer to there as “different sizes of infinity” are different cardinalities of sets. As you note, what sorts of infinities you have to use depends on what you are trying to measure; raw cardinalities are rarely the right notion of size, here we want to think in a measure-theoretic context. But it’s worth noting that for measuring other things different systems of infinite numbers must be used; cardinalities and “infinities” should not be identified.
Huh, can you define an improper uniform distribution over the integers like you can occasionally for the real line? Or does that always lead to an improper posterior?
The question is not which set is larger, which is in any case almost meaningless since both are infinite, but which set has larger probability measure.
I do think it’s meaningful to talk about different sizes of infinity (for example, countable vs. uncountable), but probability measure is probably more relevant.
To expand on that point—what you are refer to there as “different sizes of infinity” are different cardinalities of sets. As you note, what sorts of infinities you have to use depends on what you are trying to measure; raw cardinalities are rarely the right notion of size, here we want to think in a measure-theoretic context. But it’s worth noting that for measuring other things different systems of infinite numbers must be used; cardinalities and “infinities” should not be identified.
-
In some cases, there’s no way to define a uniform distribution (i.e. over the integers), so you’ve got to do something else.
Huh, can you define an improper uniform distribution over the integers like you can occasionally for the real line? Or does that always lead to an improper posterior?