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February 2021 Bayesian graph selection consistency under model misspecification
Yabo Niu, Debdeep Pati, Bani K. Mallick
Bernoulli 27(1): 637-672 (February 2021). DOI: 10.3150/20-BEJ1253


Gaussian graphical models are a popular tool to learn the dependence structure in the form of a graph among variables of interest. Bayesian methods have gained in popularity in the last two decades due to their ability to simultaneously learn the covariance and the graph. There is a wide variety of model-based methods to learn the underlying graph assuming various forms of the graphical structure. Although for scalability of the Markov chain Monte Carlo algorithms, decomposability is commonly imposed on the graph space, its possible implication on the posterior distribution of the graph is not clear. An open problem in Bayesian decomposable structure learning is whether the posterior distribution is able to select a meaningful decomposable graph that is “close” to the true non-decomposable graph, when the dimension of the variables increases with the sample size. In this article, we explore specific conditions on the true precision matrix and the graph, which results in an affirmative answer to this question with a commonly used hyper-inverse Wishart prior on the covariance matrix and a suitable complexity prior on the graph space. In absence of structural sparsity assumptions, our strong selection consistency holds in a high-dimensional setting where $p=O(n^{\alpha })$ for $\alpha <1/3$. We show when the true graph is non-decomposable, the posterior distribution concentrates on a set of graphs that are minimal triangulations of the true graph.


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Yabo Niu. Debdeep Pati. Bani K. Mallick. "Bayesian graph selection consistency under model misspecification." Bernoulli 27 (1) 637 - 672, February 2021.


Received: 1 March 2020; Revised: 1 July 2020; Published: February 2021
First available in Project Euclid: 20 November 2020

zbMATH: 07282865
MathSciNet: MR4177384
Digital Object Identifier: 10.3150/20-BEJ1253

Keywords: decomposable graph , Gaussian graphical model , graph selection consistency , Hyper-Inverse Wishart Distribution , Minimal triangulation , model misspecification , partial correlation

Rights: Copyright © 2021 Bernoulli Society for Mathematical Statistics and Probability


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Vol.27 • No. 1 • February 2021
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