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December 2019 Minimax posterior convergence rates and model selection consistency in high-dimensional DAG models based on sparse Cholesky factors
Kyoungjae Lee, Jaeyong Lee, Lizhen Lin
Ann. Statist. 47(6): 3413-3437 (December 2019). DOI: 10.1214/18-AOS1783

Abstract

In this paper we study the high-dimensional sparse directed acyclic graph (DAG) models under the empirical sparse Cholesky prior. Among our results, strong model selection consistency or graph selection consistency is obtained under more general conditions than those in the existing literature. Compared to Cao, Khare and Ghosh [Ann. Statist. (2019) 47 319–348], the required conditions are weakened in terms of the dimensionality, sparsity and lower bound of the nonzero elements in the Cholesky factor. Furthermore, our result does not require the irrepresentable condition, which is necessary for Lasso-type methods. We also derive the posterior convergence rates for precision matrices and Cholesky factors with respect to various matrix norms. The obtained posterior convergence rates are the fastest among those of the existing Bayesian approaches. In particular, we prove that our posterior convergence rates for Cholesky factors are the minimax or at least nearly minimax depending on the relative size of true sparseness for the entire dimension. The simulation study confirms that the proposed method outperforms the competing methods.

Citation

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Kyoungjae Lee. Jaeyong Lee. Lizhen Lin. "Minimax posterior convergence rates and model selection consistency in high-dimensional DAG models based on sparse Cholesky factors." Ann. Statist. 47 (6) 3413 - 3437, December 2019. https://doi.org/10.1214/18-AOS1783

Information

Received: 1 February 2018; Revised: 1 October 2018; Published: December 2019
First available in Project Euclid: 31 October 2019

Digital Object Identifier: 10.1214/18-AOS1783

Subjects:
Primary: 62C20
Secondary: 62C12, 62F15

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 6 • December 2019
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