  In latent Gaussian trees the pairwise correlation signs between the variables
are intrinsically unrecoverable. 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 achieve two fundamental
goals: First, we quantify the amount of information loss due to unrecoverable
sign information. Second, we show the importance of such information in
determining the maximum achievable rate region, in which the observed output
vector can be synthesized, given its probability density function. In
particular, we model the graphical model as a communication channel and propose
a new layered encoding framework to synthesize observed data using upper layer
Gaussian inputs and independent Bernoulli correlation sign inputs from each
layer. We find the achievable rate region for the rate tuples of multi-layer
latent Gaussian messages to synthesize the desired observables.
