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Abstract:
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A main question in graphical models and causal inference is that given a probability distribution P (which is usually an underlying distribution of data), whether there is a graph (or graphs) to which P is faithful. Here we provide a theoretical solution to this problem as well as an algorithm to generate such graphs if they exist. In order to do so, we exploit a generalization of ordering, called preordering, of the nodes of the (mixed) graphs. This allows us to provide sufficient conditions for a given probability distribution to be Markov to a graph with the minimum possible number of edges, and more importantly, necessary and sufficient conditions for a given probability distribution to be faithful to a graph. We then introduce an algorithm for selecting graphical models based on the results for faithfulness.
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