The correlation space of Gaussian latent tree models and model selection without fitting

Abstract

We provide a complete description of possible covariance matrices consistent with a Gaussian latent tree model for any tree. We then present techniques for utilising these constraints to assess whether observed data is compatible with that Gaussian latent tree model. Our method does not require us first to fit such a tree. We demonstrate the usefulness of the inverse-Wishart distribution for performing preliminary assessments of tree-compatibility using semialgebraic constraints. Using results from Drton et al. (2008) we then provide the appropriate moments required for test statistics for assessing adherence to these equality constraints. These are shown to be effective even for small sample sizes and can be easily adjusted to test either the entire model or only certain macrostructures hypothesized within the tree. We illustrate our exploratory tetrad analysis using a linguistic application and our confirmatory tetrad analysis using a biological application.

Synthesis of Gaussian trees with correlation sign ambiguity: An information theoretic approach

“…However, we show that to obtain the maximum rate region to synthesize the output, one may minimize I(X; Y), which in turn will be equivalent to maximizing the conditional mutual information I(X; B|Y), hence, showing the maximum amount of lost sign information. In such settings, we show that the input B and the output X are independent, by which we provide another reason on why previous learning approaches [1,3] are incapable of inferring the sign information.…”

“…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.…”

Successive synthesis of latent Gaussian trees

“…They have diverse applications in social networks, biology, and economics [1], [2], to name a few. Gaussian trees in particular have attracted much attention [2] due to their sparse structures, as well as existing computationally efficient algorithms in learning the underlying topologies [3], [4]. In this paper we assume that the parameters and structure information of the latent Gaussian tree is provided.…”

Maximum likelihood estimation in Gaussian models under total positivity

“…Signed MTP 2 distributions are relevant, for example, because of their appearance when studying tree models. Gaussian tree models with hidden variables have many applications, in particular related to modeling evolutionary processes; see, e.g., [7,36]. As an important submodel they contain the Brownian motion tree model [14].…”