The Annals of Statistics

Combinatorial inference for graphical models

Matey Neykov, Junwei Lu, and Han Liu

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We propose a new family of combinatorial inference problems for graphical models. Unlike classical statistical inference where the main interest is point estimation or parameter testing, combinatorial inference aims at testing the global structure of the underlying graph. Examples include testing the graph connectivity, the presence of a cycle of certain size, or the maximum degree of the graph. To begin with, we study the information-theoretic limits of a large family of combinatorial inference problems. We propose new concepts including structural packing and buffer entropies to characterize how the complexity of combinatorial graph structures impacts the corresponding minimax lower bounds. On the other hand, we propose a family of novel and practical structural testing algorithms to match the lower bounds. We provide numerical results on both synthetic graphical models and brain networks to illustrate the usefulness of these proposed methods.

Article information

Ann. Statist., Volume 47, Number 2 (2019), 795-827.

Received: August 2016
Revised: August 2017
First available in Project Euclid: 11 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F03: Hypothesis testing 62F04 62H15: Hypothesis testing

Graph structural inference minimax testing uncertainty assessment multiple hypothesis testing post-regularization inference


Neykov, Matey; Lu, Junwei; Liu, Han. Combinatorial inference for graphical models. Ann. Statist. 47 (2019), no. 2, 795--827. doi:10.1214/17-AOS1650.

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Supplemental materials

  • Supplement to “Combinatorial inference for graphical models”. The Supplementary Material contains proofs and derivations of some of the main results of the paper, as well as simulation results and real data analysis.