Locally associated graphical models and mixed convex exponential families

TL;DR

This paper introduces local association in graphical models, explores mixed convex exponential families, and develops algorithms like positive graphical lasso and GOLAZO for efficient estimation in gene expression data analysis.

Contribution

It proposes the concept of local association, studies mixed convex exponential families, and introduces the GOLAZO algorithm for graphical model estimation.

Findings

Mixed dual likelihood estimator has exact and asymptotic properties.

Positive graphical lasso effectively relaxes positivity constraints.

GOLAZO algorithm ensures the existence of optima in estimation problems.

Abstract

The notion of multivariate total positivity has proved to be useful in finance and psychology but may be too restrictive in other applications. In this paper we propose a concept of local association, where highly connected components in a graphical model are positively associated and study its properties. Our main motivation comes from gene expression data, where graphical models have become a popular exploratory tool. The models are instances of what we term mixed convex exponential families and we show that a mixed dual likelihood estimator has simple exact properties for such families as well as asymptotic properties similar to the maximum likelihood estimator. We further relax the positivity assumption by penalizing negative partial correlations in what we term the positive graphical lasso. Finally, we develop a GOLAZO algorithm based on block-coordinate descent that applies to a number of optimization procedures that arise in the context of graphical models, including the estimation problems described above. We derive results on existence of the optimum for such problems.