^{1,2}

^{2}

^{1}

^{2}

Graphical models are useful for characterizing conditional and marginal independence structures in high-dimensional distributions. An important class of graphical models is covariance graph models, where the nodes of a graph represent different components of a random vector, and the absence of an edge between any pair of variables implies marginal independence. Covariance graph models also represent more complex conditional independence relationships between subsets of variables. When the covariance graph captures or reflects all the conditional independence statements present in the probability distribution, the latter is said to be

Markov random fields and graphical models are widely used to represent conditional independences in a given multivariate probability distribution (see [

The

Note that the subscript zero is invoked for covariance graphs (i.e.,

Both concentration and covariance graphs not only are used to encode pairwise relationships between pairs of variables in the random vector

The multivariate distribution

We will call the

We first define the concept of separation on graphs. Let

The outline of this paper is as follows. Section

This section introduces notation and terminology that is required in subsequent sections. An undirected graph

Let

For a connected graph, a

In this section, we present a brief overview of concentration and covariance graphs in the case when the probability distribution

As seen earlier in (

We now proceed to study the faithfulness assumption in the context of multivariate Gaussian distributions and when the associated covariance graphs are trees. The main result of this paper, presented in Theorem

Let

The proof of Theorem

Let

The lemma above follows immediately from a basic result in linear algebra which gives the cofactor expression for the inverse of a square matrix. In particular, for an invertible matrix

A simple proof can be found in Brualdi and Cvetkovic [

The proof of our main theorem (Theorem

Let

if a given connected component in

Proof of (i): the fact that

Proof of (ii): let us assume now that the covariance graph

We now give an example illustrating the main result in this paper (Theorem

Consider a Gaussian random vector

Consider the sets

Let us now examine the relationship between

Since 2 and 5 are connected by exactly one path in

Now, recall that, for any Gaussian random vector

Hence, we conclude that since

Covariance graph in Example

An 8-vertex covariance tree

Sub-graph

We now proceed to the proof of Theorem

Without loss of generality, we assume that

As

In this case, the entire path between

In this case, there exists a pair of vertices

Let us now use Lemma

The dual result of the theorem above for the case of concentration trees was proved by Becker et al. [

In this note we looked at the class of multivariate Gaussian distributions that are Markov with respect to covariance graphs and prove that Gaussian distributions which have trees as their covariance graphs are necessarily faithful. The method of proof used in the paper is also vastly different in nature from the proof of the analogous result for concentration graph models. Hence, the approach that is used could potentially have further implications. Future research in this area will explore if the analysis presented in this paper can be extended to other classes of graphs or distributions.

D. Malouche was supported in part by a Fullbright Fellowship Grant 68434144. B. Rajaratnam was supported in part by NSF grants DMS0906392, DMS(CMG)1025465, AGS1003823, NSA H98230-11-1-0194, and SUFSC10-SUSHSTF09SMSCVISG0906.