Segregated Graphs and Marginals of Chain Graph Models
Bayesian networks are a popular representation of asymmetric (for example causal) relationships between random variables. Markov random fields (MRFs) are a complementary model of symmetric relationships used in computer vision, spatial modeling, and social and gene expression networks. A chain graph model under the Lauritzen-Wermuth-Frydenberg interpretation (hereafter a chain graph model) generalizes both Bayesian networks and MRFs, and can represent asymmetric and symmetric relationships together. As in other graphical models, the set of marginals from distributions in a chain graph model induced by the presence of hidden variables forms a complex model. One recent approach to the study of marginal graphical models is to consider a well-behaved supermodel. Such a supermodel of marginals of Bayesian networks, defined only by conditional independences, and termed the ordinary Markov model, was studied at length in [6]. In this paper, we show that special mixed graphs which we call segregated graphs can be associated, via a Markov property, with supermodels of marginals of chain graphs defined only by conditional independences. Special features of segregated graphs imply the existence of a very natural factorization for these supermodels, and imply many existing results on the chain graph model, and the ordinary Markov model carry over. Our results suggest that segregated graphs define an analogue of the ordinary Markov model for marginals of chain graph models. We illustrate the utility of segregated graphs for analyzing outcome interference in causal inference via simulated datasets.
The authors try to tackle the problem when bidirected edge (<->) and undirected edge (-) meet in a loopless mixed graph. They propose to construct a new mixed graph called segregated graph (SG) (where bidirected edge (<->) and undirected edge (-) never meet) which preserves all conditional independencies on the observed variables. I like the originality of the idea. I am somewhat disappointed by the clarity of the paper, which affects the quality too. Also the significance of the problem considered is not well addressed. Here I give more details on each criterion: *Clarity: I like the authors' explanation of the big picture before going to details, which helps to clarify to some degree. However, once we check the details, it becomes less clear caused by typos and undefined/undeclared symbols. It would be much easier to read if the authors follow notations in any of the well known (and recent) publications on this topic such as [4], [1*], and [2*].*Quality: it is dragged down by the level of clarity. Also I notice that if we move most of the proofs to the supplementary file, the paper would have much more spare space (2-3 pages perhaps). This suggests much more content can be added to the paper. *Originality: good. *Significance: the importance of the problem is only lightly touched (line 128-129) with a bit further discussion on the implication in conclusions. Missing references: [1*] Sadeghi, K. and Lauritzen, S. (2014). Markov properties for mixed graphs. Bernoulli 20, 676-696. MR3178514 [2*] Wermuth, N. (2011). Probability distributions with summary graph structure. Bernoulli 17, 845-879. MR2817608After reviewers' dissuasion and seeing authors' response, I've increased the score by one step.