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.
Funding Statement
The second author was supported in part by Ayudas Fundación BBVA a Proyectos de Investigación Científica en Matemáticas 2022.
Acknowledgments
We are grateful to Robert Castelo for providing us with an interesting dataset and for patiently explaining the underlying biology to us. We would also like the referees for helpful comments and for pointing out a mistake in an earlier version of the paper.
Citation
Steffen Lauritzen. Piotr Zwiernik. "Locally associated graphical models and mixed convex exponential families." Ann. Statist. 50 (5) 3009 - 3038, October 2022. https://doi.org/10.1214/22-AOS2219
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