I think the basic CCT theorem is wrong. Consider a game where there is a coin that will be flipped, and you are going to predict the probability that this coin comes up heads. You will be scored using a proper scoring rule, such as square error (i.e. you get loss p^2 if it’s heads and (1-p)^2 if it’s tails). The policy of always saying p = 0 is admissible, since no other policy is better when the coin always comes up tails. But it is not the result of any non-dogmatic prior (which would say a probability strictly between 0 and 1).
I didn’t understand your proof. I get the argument for why there’s a hyperplane but can’t the hyperplane be parallel to one of the axes, so it never intersects that one?
Ah, yeah, you’re right. The separating hyperplane theorem only gives us ≤, and I was assuming <.
I think “admissible if and only if non-dogmatic” may still hold as I stated it, because I don’t see how to set up an example like the one you give when A is finite. I’m editing the post anyway, since (1) I don’t know how to how that at the moment, and (2) the if and only if falls apart for infinite action sets as in your example anyway, which makes it kind of feel “wrong in spirit”.
I think the basic CCT theorem is wrong. Consider a game where there is a coin that will be flipped, and you are going to predict the probability that this coin comes up heads. You will be scored using a proper scoring rule, such as square error (i.e. you get loss p^2 if it’s heads and (1-p)^2 if it’s tails). The policy of always saying p = 0 is admissible, since no other policy is better when the coin always comes up tails. But it is not the result of any non-dogmatic prior (which would say a probability strictly between 0 and 1).
I didn’t understand your proof. I get the argument for why there’s a hyperplane but can’t the hyperplane be parallel to one of the axes, so it never intersects that one?
Ah, yeah, you’re right. The separating hyperplane theorem only gives us ≤, and I was assuming <.
I think “admissible if and only if non-dogmatic” may still hold as I stated it, because I don’t see how to set up an example like the one you give when A is finite. I’m editing the post anyway, since (1) I don’t know how to how that at the moment, and (2) the if and only if falls apart for infinite action sets as in your example anyway, which makes it kind of feel “wrong in spirit”.