If the functions F1 F2 etc. originally came from a Bayesian network, won’t this recover that precise network?
I think this is right, if you know that the factors were learned by fitting them to a Bayesian network, you can recover what that network must have been. And you can go even further, if you only have a joint distribution you can use the techniques of the original article to see which Bayesian networks could be consistent with it.
But there is a separate question about why we are interested in Bayesian networks in the first place. SilasBarta seemed to claim that you are naturally led to them if you are interested in representing probability distributions efficiently. But for that purpose (I claim), you only need the idea of factors, not the directed graph structure. E.g. a probability distribution which fits the (equivalent) Bayesian networks A → B → C or A ← B ← C or A ← B → C can be efficiently represented as F1(a,b) F2(b,c). You would not think of representing it as F1(a) F2(a,b) F3(b,c) unless you were already interested in causality.
I think this is right, if you know that the factors were learned by fitting them to a Bayesian network, you can recover what that network must have been. And you can go even further, if you only have a joint distribution you can use the techniques of the original article to see which Bayesian networks could be consistent with it.
But there is a separate question about why we are interested in Bayesian networks in the first place. SilasBarta seemed to claim that you are naturally led to them if you are interested in representing probability distributions efficiently. But for that purpose (I claim), you only need the idea of factors, not the directed graph structure. E.g. a probability distribution which fits the (equivalent) Bayesian networks A → B → C or A ← B ← C or A ← B → C can be efficiently represented as F1(a,b) F2(b,c). You would not think of representing it as F1(a) F2(a,b) F3(b,c) unless you were already interested in causality.