Isn’t all the information you care about supposed to be encapsulated in your probability distribution?
No. As another (yours is one) simple counterexample, if I flip a fair coin 100 times you expect around 50 heads, but if I either choose a double-head or double-tail coin and flip that 100 times, you expect either 100 heads or 100 tails—and yet the probability of the first flip is still 50⁄50.
A distribution over models solves this problem. IIRC you don’t have to regress further, but I don’t remember where (or even if) I saw that result.
but if I either choose a double-head or double-tail coin and flip that 100 times,
To clarify: if you know Guy chose either a double-head or double-tail coin, but you have no idea which, then you should assign 50% to heads on the first flip, then either 0% or 100% to heads after, since you’ll the know which one it was.
It’s been linked too often already in this thread, but the example in Priors as Mathematical Objects neatly demonstrates how a prior is more than just a probability distribution, and how Simetrical’s question doesn’t lead to paradox.
No. As another (yours is one) simple counterexample, if I flip a fair coin 100 times you expect around 50 heads, but if I either choose a double-head or double-tail coin and flip that 100 times, you expect either 100 heads or 100 tails—and yet the probability of the first flip is still 50⁄50.
A distribution over models solves this problem. IIRC you don’t have to regress further, but I don’t remember where (or even if) I saw that result.
To clarify: if you know Guy chose either a double-head or double-tail coin, but you have no idea which, then you should assign 50% to heads on the first flip, then either 0% or 100% to heads after, since you’ll the know which one it was.
It’s been linked too often already in this thread, but the example in Priors as Mathematical Objects neatly demonstrates how a prior is more than just a probability distribution, and how Simetrical’s question doesn’t lead to paradox.