What if X is something that only 𝔼 knows, like a random number that the agent that 𝔼 represents just thought up? Then there would be no way for this equality to hold? Maybe it should be something like this instead?
𝔼[X|𝔼¹[X]] = 𝔼[X|𝔼¹[X], Φ(𝔼¹, 𝔼²)]
I’m not sure this is right either, because if 𝔼 is smart enough maybe it can extract some information about X from Φ(𝔼¹, 𝔼²) that’s not in 𝔼¹[X]. Edit: Never mind, that’s fine because it just means that 𝔼¹ has to be smart enough that 𝔼¹[X] takes into account everything that 𝔼 might be able to extract from Φ(𝔼¹, 𝔼²).
In order to satisfy this definition, 𝔼¹ needs to know every particular fact 𝔼 knows. It would be nice to have a definition that got at the heart of the matter while relaxing this requirement.
I don’t think your condition gets around this requirement. Suppose that Y is a bit that 𝔼 knows and 𝔼¹ does not, Z[0] and Z[1] are two hard-to-estimate quantities (that 𝔼¹ and 𝔼² know but 𝔼 does not), and that X=Z[Y].
What if X is something that only 𝔼 knows, like a random number that the agent that 𝔼 represents just thought up? Then there would be no way for this equality to hold? Maybe it should be something like this instead?
𝔼[X|𝔼¹[X]] = 𝔼[X|𝔼¹[X], Φ(𝔼¹, 𝔼²)]
I’m not sure this is right either, because if 𝔼 is smart enough maybe it can extract some information about X from Φ(𝔼¹, 𝔼²) that’s not in 𝔼¹[X]. Edit: Never mind, that’s fine because it just means that 𝔼¹ has to be smart enough that 𝔼¹[X] takes into account everything that 𝔼 might be able to extract from Φ(𝔼¹, 𝔼²).
In order to satisfy this definition, 𝔼¹ needs to know every particular fact 𝔼 knows. It would be nice to have a definition that got at the heart of the matter while relaxing this requirement.
I don’t think your condition gets around this requirement. Suppose that Y is a bit that 𝔼 knows and 𝔼¹ does not, Z[0] and Z[1] are two hard-to-estimate quantities (that 𝔼¹ and 𝔼² know but 𝔼 does not), and that X=Z[Y].