The Annals of Statistics

The maximum likelihood threshold of a path diagram

Mathias Drton, Christopher Fox, Andreas Käufl, and Guillaume Pouliot

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Linear structural equation models postulate noisy linear relationships between variables of interest. Each model corresponds to a path diagram, which is a mixed graph with directed edges that encode the domains of the linear functions and bidirected edges that indicate possible correlations among noise terms. Using this graphical representation, we determine the maximum likelihood threshold, that is, the minimum sample size at which the likelihood function of a Gaussian structural equation model is almost surely bounded. Our result allows the model to have feedback loops and is based on decomposing the path diagram with respect to the connected components of its bidirected part. We also prove that if the sample size is below the threshold, then the likelihood function is almost surely unbounded. Our work clarifies, in particular, that standard likelihood inference is applicable to sparse high-dimensional models even if they feature feedback loops.

Article information

Ann. Statist., Volume 47, Number 3 (2019), 1536-1553.

Received: February 2018
First available in Project Euclid: 13 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation 62J05: Linear regression

Covariance matrix graphical model maximum likelihood normal distribution path diagram structural equation model


Drton, Mathias; Fox, Christopher; Käufl, Andreas; Pouliot, Guillaume. The maximum likelihood threshold of a path diagram. Ann. Statist. 47 (2019), no. 3, 1536--1553. doi:10.1214/18-AOS1724.

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