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

Latent Gaussian tree model learning algorithms lack in fully recovering the sign information regarding pairwise correlation values between variables. Such information is vital since it completely determines the direction in which two variables are associated. In this work, we resort to information theoretical approaches to quantify information in regard to lost correlation signs in the recovered model. %We model the graphical model as a communication channel and apply the tools in information theory to model the lost sign information. We model the graphical model as a communication channel and propose a new layered encoding framework to synthesize observed data using top layer Gaussian inputs and independent Bernoulli correlation sign inputs from each layer. We show that by maximizing the inferred information about the correlation sign information, one may also find the largest achievable rate region for the rate tuples of multi-layer latent Gaussian messages and missing correlation sign messages to synthesize the desired observables.