On a hypergraph probabilistic graphical model
This work provides a more expressive and computationally efficient probabilistic graphical model for researchers in machine learning and statistics, though it appears incremental as it builds upon existing chain graph frameworks.
The authors tackled the problem of modeling complex probabilistic dependencies and causal interactions by proposing Bayesian hypergraphs, a directed acyclic hypergraph framework that offers finer factorizations and more efficient computational procedures than existing models like Bayesian networks or chain graphs, with proven equivalence of Markov properties and extended causal interpretation.
We propose a directed acyclic hypergraph framework for a probabilistic graphical model that we call Bayesian hypergraphs. The space of directed acyclic hypergraphs is much larger than the space of chain graphs. Hence Bayesian hypergraphs can model much finer factorizations than Bayesian networks or LWF chain graphs and provide simpler and more computationally efficient procedures for factorizations and interventions. Bayesian hypergraphs also allow a modeler to represent causal patterns of interaction such as Noisy-OR graphically (without additional annotations). We introduce global, local and pairwise Markov properties of Bayesian hypergraphs and prove under which conditions they are equivalent. We define a projection operator, called shadow, that maps Bayesian hypergraphs to chain graphs, and show that the Markov properties of a Bayesian hypergraph are equivalent to those of its corresponding chain graph. We extend the causal interpretation of LWF chain graphs to Bayesian hypergraphs and provide corresponding formulas and a graphical criterion for intervention.