A form of the anthropic principle is as follows: "We can observe the universe only because we can exist within it in some way such that we can observe it, and it exists such that we can observe it."

What mathematical consequence does this have? I know it's broadly a problem of Bayesian probability, and we must consider all that we see from the perspective P(A|B), A = some aspect of observed reality, B = we think, therefore we are.

Can this be formulated in some useful and general way to answer questions about the universe, existential, cosmological or otherwise, or do the mathematics here give us little information?

NOTE: I know that the anthropic principle is often stated in a much more specific way and looked at from the perspective of cosmology, but that's not what I'm looking for here.

Edit: To clarify the mathematical content of this question I'll give two examples (one from a comment below).

1) I've seen claims like "the anthropic principle indicates that we most likely live at a time such that half of all people that have ever been born have been born". I want to know if a statement like this is at all reasonable or not.

2) Consider it in these (not entirely sufficient) terms: You have a vague outline of a set of prior distributions in addition to some error-prone observations whose errors depend on the prior distribution. How can you glean information about the prior distribution.