Now to answer our big question from the previous section: I can find some Λ∗ satisfying the conditions exactly when all of the Xi’s are independent given the “perfectly redundant” information. In that case, I just set Λ∗ to be exactly the quantities conserved under the resampling process, i.e. the perfectly redundant information itself.
In the original post on redundant information, I didn’t find a definition for the “quantities conserved under the resampling process”. You name this F(X) in that post.
Just to be sure: is your claim that if F(X) exists that contains exactly the conserved quantities and nothing else, then you can define Λ∗ like this? Or is the claim even stronger and you think such F can always be constructed?
Edit: Flagging that I now think this comment is confused. One can simply defineF(X)=P(X∞∣X) as the conditional, which is a composition of the random variable X and the function F:x↦P(X∞∣X=x)
In the original post on redundant information, I didn’t find a definition for the “quantities conserved under the resampling process”. You name this F(X) in that post.
Just to be sure: is your claim that if F(X) exists that contains exactly the conserved quantities and nothing else, then you can define Λ∗ like this? Or is the claim even stronger and you think such F can always be constructed?
Edit: Flagging that I now think this comment is confused. One can simply define F(X)=P(X∞∣X) as the conditional, which is a composition of the random variable X and the function F:x↦P(X∞∣X=x)