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**1 - 4**of**4**### Every LWF and AMP Chain Graph Originates From a Set of Causal Models. ArXiv e-prints

, 2013

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### Error AMP Chain Graphs

, 2013

"... Any regular Gaussian probability distribution that can be represented by an AMP chain graph (CG) can be expressed as a system of linear equations with correlated errors whose structure depends on the CG. However, the CG represents the errors implicitly, as no nodes in the CG correspond to the erro ..."

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Any regular Gaussian probability distribution that can be represented by an AMP chain graph (CG) can be expressed as a system of linear equations with correlated errors whose structure depends on the CG. However, the CG represents the errors implicitly, as no nodes in the CG correspond to the errors. We propose in this paper to add some deterministic nodes to the CG in order to represent the errors explicitly. We call the result an EAMP CG. We will show that, as desired, every AMP CG is Markov equivalent to its corresponding EAMP CG under marginalization of the error nodes. We will also show that every EAMP CG under marginalization of the error nodes is Markov equivalent to some LWF CG under marginalization of the error nodes, and that the latter is Markov equivalent to some directed and acyclic graph (DAG) under marginalization of the error nodes and conditioning on some selection nodes. This is important because it implies that the independence model represented by an AMP CG can be accounted for by some data generating process that is partially observed and has selection bias. Finally, we will show that EAMP CGs are closed under marginalization. This is a desirable feature because it guarantees parsimonious models under marginalization.

### Approximate Counting of Graphical Models Via MCMC Revisited

"... Abstract. We apply MCMC sampling to approximately calculate some quantities, and discuss their implications for learning directed and acyclic graphs (DAGs) from data. Specifically, we calculate the approximate ratio of essential graphs (EGs) to DAGs for up to 31 nodes. Our ratios suggest that the av ..."

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Abstract. We apply MCMC sampling to approximately calculate some quantities, and discuss their implications for learning directed and acyclic graphs (DAGs) from data. Specifically, we calculate the approximate ratio of essential graphs (EGs) to DAGs for up to 31 nodes. Our ratios suggest that the average Markov equivalence class is small. We show that a large majority of the classes seem to have a size that is close to the average size. This suggests that one should not expect more than a moderate gain in efficiency when searching the space of EGs instead of the space of DAGs. We also calculate the approximate ratio of connected EGs to connected DAGs, of connected EGs to EGs, and of connected DAGs to DAGs. These new ratios are interesting because, as we will see, the DAG or EG learnt from some given data is likely to be connected. Furthermore, we prove that the latter ratio is asymptotically 1. Finally, we calculate the approximate ratio of EGs to largest chain graphs for up to 25 nodes. Our ratios suggest that Lauritzen-Wermuth-Frydenberg chain graphs are considerably more expressive than DAGs. We also re-port similar approximate ratios and conclusions for multivariate regres-sion chain graphs. 1

### Learning Marginal AMP Chain Graphs under Faithfulness

, 2014

"... Marginal AMP chain graphs are a recently introduced family of models that is based on graphs that may have undirected, directed and bidirected edges. They unify and generalize the AMP and the multivariate regression interpretations of chain graphs. In this paper, we present a constraint based algo ..."

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Marginal AMP chain graphs are a recently introduced family of models that is based on graphs that may have undirected, directed and bidirected edges. They unify and generalize the AMP and the multivariate regression interpretations of chain graphs. In this paper, we present a constraint based algorithm for learning a marginal AMP chain graph from a probability distribution which is faithful to it. We also show that the extension of Meek’s conjecture to marginal AMP chain graphs does not hold, which compromises the development of efficient and correct score+search learning algorithms under assumptions weaker than faithfulness.