Thank you for the explanation. We need some better notation for this: write I(x) to mean that I observed x (or will observe it). Write E(x) to mean that I know that there exists with certainty someone who observed x (or will observe it).
First, in general, we don’t expect P(I(x)) = P(E(x)). For example, P(I(room 1)|I(room 2)) = 0 but P(E(room 1) | I(room 2)) = 1.
You define “simple Bayes” to mean:
P(y | I(x)) = P(y | E(x)) for any y and x
I would argue that you really need to pick a totally different name for this property since it doesn’t have anything to do with Bayes’ law. (If anything, I would call it “nonstandard Bayes”.) And your definition of it in your article needs to be clearer: as stated, it’s just about I(x) without mentioning E(x).
This is also the same crux that the paradox has always revolved around: does finding out who you are give you information? I think that’s closer to the standard phrasing and makes it more clear what you’re being asked to give up or not. You’re not being asked to give up Bayes’ law: that’s always true.
Thank you for the explanation. We need some better notation for this: write I(x) to mean that I observed x (or will observe it). Write E(x) to mean that I know that there exists with certainty someone who observed x (or will observe it).
First, in general, we don’t expect P(I(x)) = P(E(x)). For example, P(I(room 1)|I(room 2)) = 0 but P(E(room 1) | I(room 2)) = 1.
You define “simple Bayes” to mean:
P(y | I(x)) = P(y | E(x)) for any y and x
I would argue that you really need to pick a totally different name for this property since it doesn’t have anything to do with Bayes’ law. (If anything, I would call it “nonstandard Bayes”.) And your definition of it in your article needs to be clearer: as stated, it’s just about I(x) without mentioning E(x).
This is also the same crux that the paradox has always revolved around: does finding out who you are give you information? I think that’s closer to the standard phrasing and makes it more clear what you’re being asked to give up or not. You’re not being asked to give up Bayes’ law: that’s always true.