Testing Unfaithful Gaussian Graphical Models
The global Markov property for Gaussian graphical models ensures graph separation implies conditional independence. Specifically if a node set S graph separates nodes u and v then X u is conditionally independent of X v given X S . The opposite direction need not be true, that is, X u ⊥ X v | X S need not imply S is a node separator of u and v. When it does, the relation X u ⊥ X v | X S is called faithful. In this paper we provide a characterization of faithful relations and then provide an algorithm to test faithfulness based only on knowledge of other conditional relations of the form X i ⊥ X j | X S .
Identifying conditional independence of variables in graphical models is a key to finding tractable solutions, and faithfulness is a key condition of these relationships. The authors provide necessary and sufficient conditions for determining faithfulness in Gaussian graphical models (based on partitioning variables outside the conditioning set into two disjoint subsets), and show how this theoretical result can be translated into an algorithm for determining if a distribution is faithful.PROS:- Clear, well-written paper with illustrative examples- Addresses a relevant problem and provides a meaningful theoretical result - Provides a practical test for faithfulness in Gaussian graphical modelCONS:- Theoretical result is restricted to Gaussian graphical modelsQuality: This paper provides addresses a theoretical problem  (faithfulness in Gaussian graphical models). The claims are well-reasoned and the proofs support the claims. The resulting algorithm provides a useful test of faithfulness. Although the claims are extremely specific, the results are well-reasoned.Clarity: This paper is very well written (although minor grammatical issues), logically organized,  and effectively introduces major concepts, illustrates ideas with examples. The proofs are clear and easy to follow.Originality: I am unfamiliar of any work that addresses this particular problem, but I do not believe the analysis is particularly innovative either.Significance: The specificity of this theoretical result (Gaussian graphical models) limits the significance of the paper, but this paper will be very useful to the particular community it targets.