Chernoff Information between Gaussian Trees

In this paper, we aim to provide a systematic study of the relationship be-tween Chernoff information and topological, as well as algebraic propertiesof the corresponding Gaussian tree graphs for the underlying graphical test-ing problems. We first show the relationship between Chernoff informationand generalized eigenvalues of the associated covariance matrices. It is thenproved that Chernoff information between two Gaussian trees sharing certainlocal subtree structures can be transformed into that of two smaller trees.Under our proposed grafting operations, bottleneck Gaussian trees, namely,Gaussian trees connected by one such operation, can thus be simplified intotwo 3-node Gaussian trees, whose topologies and edge weights are subjectto the specifics of the operation. Thereafter, we provide a thorough studyabout how Chernoff information changes when small differences are accu-mulated into bigger ones via concatenated grafting operations. It is shownthat the two Gaussian trees connected by more than one grafting operationmay not have bigger Chernoff information than that of one grafting opera-tion unless these grafting operations are separate and independent. At theend, we propose an optimal linear dimensional reduction method related togeneralized eigenvalues.