This article studies a natural and interesting mathematical question: which algebraic relations hold between Bayes nets? In other words, if a collection of random variables is consistent with several Bayes nets, what other Bayes nets does it also have to be consistent with? The question is studied both for exact consistency and for approximate consistency: in the latter case, the joint distribution is KL-close to a distribution that’s consistent with the net. The article proves several rules of this type, some of them quite non-obvious. The rules have concrete applications in the authors’ research agenda.
Some further questions that I think would be interesting to study:
Can we derive a full classification of such rules?
Is there a category-theoretic story behind the rules? Meaning, is there a type of category for which Bayes nets are something akin to string diagrams and the rules follow from the categorical axioms?
Seems right, but is there a categorical derivation of the Wentworth-Lorell rules? Maybe they can be represented as theorems of the form: given an arbitrary Markov category C, such-and-such identities between string diagrams in C imply (more) identities between string diagrams in C.
This article studies a natural and interesting mathematical question: which algebraic relations hold between Bayes nets? In other words, if a collection of random variables is consistent with several Bayes nets, what other Bayes nets does it also have to be consistent with? The question is studied both for exact consistency and for approximate consistency: in the latter case, the joint distribution is KL-close to a distribution that’s consistent with the net. The article proves several rules of this type, some of them quite non-obvious. The rules have concrete applications in the authors’ research agenda.
Some further questions that I think would be interesting to study:
Can we derive a full classification of such rules?
Is there a category-theoretic story behind the rules? Meaning, is there a type of category for which Bayes nets are something akin to string diagrams and the rules follow from the categorical axioms?
I’ve been told a Bayes net is “just” a functor from a free Cartesian category to a category of probability spaces /Markov Kernels.
Seems right, but is there a categorical derivation of the Wentworth-Lorell rules? Maybe they can be represented as theorems of the form: given an arbitrary Markov category C, such-and-such identities between string diagrams in C imply (more) identities between string diagrams in C.