Chernoff information of bottleneck Gaussian trees

In this paper, our objective is to find out the determining factors of Chernoff information in distinguishing a set of Gaussian trees. In this set, each tree can be attained via a subtree removal and grafting operation from another tree. This is equivalent to asking for the Chernoff information between the most-likely confused, i.e. “bottleneck”, Gaussian trees, as shown to be the case in ML estimated Gaussian tree graphs lately. We prove that the Chernoff information between two Gaussian trees related through a subtree removal and grafting operation is the same as that between two three-node Gaussian trees, whose topologies and edge weights are subject to the underlying graph operation. In addition, such Chernoff information is shown to be determined only by the maximum generalized eigenvalue of the two Gaussian covariance matrices. The Chernoff information of scalar Gaussian variables as a result of linear transformation (LT) of the original Gaussian vectors is also uniquely determined by the same maximum generalized eigenvalue. What is even more interesting is that after incorporating the cost of measurements into a normalized Chernoff information, Gaussian variables from LT have larger normalized Chernoff information than the one based on the original Gaussian vectors, as shown in our proved bounds.