Your model uses correlational notions like “conditional independence” to make sense of it. But I think one could perhaps come with an alternate model using causal notions?
Specifically: If two variables X and Y are correlated, then they usually are so due to confounding, because there are a lot more ways that things can be confounded than that they can be causally related. So it makes sense to assume that they are confounded.
You could approximate all of the confounders of a suitably chosen set of observable variables by postulating a new variable, which affects all of the observables. This confounder then turns into your feature axis (if continuous) or cluster (if discrete).
Your model uses correlational notions like “conditional independence” to make sense of it. But I think one could perhaps come with an alternate model using causal notions?
Specifically: If two variables X and Y are correlated, then they usually are so due to confounding, because there are a lot more ways that things can be confounded than that they can be causally related. So it makes sense to assume that they are confounded.
You could approximate all of the confounders of a suitably chosen set of observable variables by postulating a new variable, which affects all of the observables. This confounder then turns into your feature axis (if continuous) or cluster (if discrete).
This is exactly right; we can interpret the abstraction model essentially along these lines as well.