I am seeking a graphical model that is a generalization of both a Markov random field (MRF) and a Bayesian network (BN).
From the Markov random field wiki page:
A Markov network or MRF is similar to a Bayesian network in its representation of dependencies; the differences being that Bayesian networks are directed and acyclic, whereas Markov networks are undirected and may be cyclic. Thus, a Markov network can represent certain dependencies that a Bayesian network cannot (such as cyclic dependencies); on the other hand, it can't represent certain dependencies that a Bayesian network can (such as induced dependencies).
From the above description, particularly the last sentence, it appears that neither MRFs nor BNs are more general than the other.
- Is there a graphical model that encompasses both MRFs and BNs? (I think I’ve answered this, see the update below)
- What is the form of its factorization?
I believe such a graphical model will need to be directed so as to be able to model the (undirected) dependencies in a MRF (by included a directed edge in each direction). Mixed graphs also may be relevant.
I believe I've answered the first part of my question. It seems the natural generalization is that of a chain graph (also see: N. Wermuth and S. L. Lauritzen. "On substantive research hypotheses, conditional independence graphs and graphical chain models." J. R. Statist. Soc. B, 52:21–50, 1990.). The second question, on their factorization, still stands. That is, what is the definition of a Markov blanket for a chain graph (the Markov blanket for a Markov random field is the neighbor set whereas for a Bayesian network, it is the parent nodes, children nodes, and parents of the children nodes).