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# Approximate Counting of Graphical Models Via MCMC

*Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics*, PMLR 2:355-362, 2007.

#### Abstract

We apply MCMC to approximately calculate (i) the ratio of directed acyclic graph (DAG) models to DAGs for up to 20 nodes, and (ii) the fraction of chain graph (CG) models that are neither undirected graph (UG) models nor DAG models for up to 13 nodes. Our results suggest that, for the numbers of nodes considered, (i) the ratio of DAG models to DAGs is not very low, (ii) the ratio of DAG models to UG models is very high, (iii) the fraction of CG models that are neither UG models nor DAG models is rather high, and (iv) the ratio of CG models to CGs is rather low. Therefore, our results suggest that (i) when learning DAG/CG models, searching the space of DAG/CG models instead of the space of DAGs/CGs can result in a moderate/considerable gain in efficiency, and (ii) learning a CG model instead of an UG model or DAG model can result in a substantially better fit of the learning data.