Aside: if a string is natural over some strings y_1, …, y_n, then it’s also natural over any subset consisting of two or more of those strings.
Is this actually true? Intuitively, it feels wrong: a natural Λ is supposed to contain exactly the information which is mutual between the y strings. But, a subset would have more mutual information than the full set. Formally, the redundancy conditional obviously descends to subsets, but mediation seems to break, at least superficially?
The strings y_1, …, y_n are not arbitrary; no natural latent exists over most sets of strings. In order for a natural latent to exist, all the information shared between any strings must be redundant across all strings.
Is this actually true? Intuitively, it feels wrong: a natural Λ is supposed to contain exactly the information which is mutual between the y strings. But, a subset would have more mutual information than the full set. Formally, the redundancy conditional obviously descends to subsets, but mediation seems to break, at least superficially?
The strings y_1, …, y_n are not arbitrary; no natural latent exists over most sets of strings. In order for a natural latent to exist, all the information shared between any strings must be redundant across all strings.