## Faithfulness of Probability Distributions and Graphs

*Kayvan Sadeghi*; 18(148):1−29, 2017.

### Abstract

A main question in graphical models and causal inference is
whether, given a probability distribution $P$ (which is usually
an underlying distribution of data), there is a graph (or
graphs) to which $P$ is faithful. The main goal of this paper is
to provide a theoretical answer to this problem. We work with
general independence models, which contain probabilistic
independence models as a special case. We exploit a
generalization of ordering, called preordering, of the nodes of
(mixed) graphs. This allows us to provide sufficient conditions
for a given independence model 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 present our results
for the general case of mixed graphs, but specialize the
definitions and results to the better-known subclasses of
undirected (concentration) and bidirected (covariance) graphs as
well as directed acyclic graphs.

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