Yes, de Finetti’s theorem shows that if our beliefs are unchanged by exchanging members of the sequence, that’s mathematically equivalent to having some “unknown probability” that we can learn by observing the sequence.
Importantly, this is always against some particular background state of knowledge, in which our beliefs are exchangeable. We ordinary have exchangeable beliefs about coin flips, for example, but may not if we had less information (such as not knowing they’re coin flips) or more (like information about the initial conditions of each flip).
In my post on unknown probabilities, I give more detail on how they are definedc which turns out to involve a specific state of background knowledge, so they only act like a “true” probability relative to that background knowledge. And how they can be interpreted as part of a physical understanding of the situation.
Personally, rather than observing that my beliefs are exchangeable and inferring an unknown probability as a mathematical fiction, I would rather “see” the unknown probability directly in my understanding of the situation, as described in my post.
Yes, de Finetti’s theorem shows that if our beliefs are unchanged by exchanging members of the sequence, that’s mathematically equivalent to having some “unknown probability” that we can learn by observing the sequence.
Importantly, this is always against some particular background state of knowledge, in which our beliefs are exchangeable. We ordinary have exchangeable beliefs about coin flips, for example, but may not if we had less information (such as not knowing they’re coin flips) or more (like information about the initial conditions of each flip).
In my post on unknown probabilities, I give more detail on how they are definedc which turns out to involve a specific state of background knowledge, so they only act like a “true” probability relative to that background knowledge. And how they can be interpreted as part of a physical understanding of the situation.
Personally, rather than observing that my beliefs are exchangeable and inferring an unknown probability as a mathematical fiction, I would rather “see” the unknown probability directly in my understanding of the situation, as described in my post.