Thane Ruthenis comments on Natural Latents: Latent Variables Stable Across Ontologies

[Thane Ruthenis]

Just skimmed for now...

We say a latent ${\Lambda }^{\prime }$ is a “redund” over observables $X_1,...,X_n$ if and only if ${\Lambda }^{\prime }$ is fully determined by each $X_i$ individually, i.e. there exist functions $f_i$ such that ${\Lambda }^{\prime }=f_i\left(X_i\right)$ for each $i$. In the approximate case, we weaken this condition to say that the entropy $H\left({\Lambda }^{\prime }|X_i\right)\le ϵ$ for all $i$, for some approximation error $ϵ$.

I see your latest result has allowed you to streamline the definitions of redundancy and redunds.

I think attempting to require ${ϵ}_{\text{red}}$ to be small in terms of ${ϵ}_{{\text{red}}^{\prime }}$ would still run into my counterexample, right? (Setups could be constructed such that requiring ${ϵ}_{\text{red}}$ to be ${ϵ}_{{\text{red}}^{\prime }}$-small would cause ${ϵ}_{\text{med}}$ to scale arbitrarily with $i$, and vice versa. So in the general case, there may exist valid redunds with the redundancy error ${ϵ}_{{\text{red}}^{\prime }}$ such that the maximal redund’s ${ϵ}_{\text{red}}$ and ${ϵ}_{\text{med}}$ (and therefore ${ϵ}_{\text{med}}+2{ϵ}_{{\text{red}}^{\prime }}$) cannot both be ${ϵ}_{{\text{red}}^{\prime }}$-small.)