Synthesis of Gaussian Trees with Correlation Sign Ambiguity: An Information Theoretic Approach

The goal of any inference algorithm is to recover the hidden parameters related to those k hidden nodes (k may be unknown). Consider a special subset of graphical models, known as latent Gaussian trees, in which the underlying structure is a tree and the joint density of the variables is captured by a Gaussian density. The Gaussian graphical models are widely studied in the literature because of a direct correspondence between conditional independence relations occurring in the model with zeros in the inverse of covariance matrix, known as the concentration matrix. There are several works such as [1,2] that have proposed efficient algorithms to infer the latent Gaussian tree parameters. In fact, Choi et al., proposed a new recursive grouping (RG) algorithm along with its improved version, i.e., Chow-Liu RG (CLRG) algorithm to recover a latent Gaussian tree that is both structural and risk consistent [1], hence it recovers the correct value for the latent parameters. They introduced a tree metric as the negative log of the absolute value of pairwise correlations to perform the algorithm. Also, Shiers et al., in [3], characterized the correlation space of latent Gaussian trees and showed the necessary and sufficient conditions under which the correlation space represents a particular latent Gaussian tree. Note that the RG algorithm can be directly related to correlation space of latent Gaussian trees in a sense that it recursively checks certain constraints on correlations to converge to a latent tree with true parameters.