The Annals of Statistics

Computation of maximum likelihood estimates in cyclic structural equation models

Mathias Drton, Christopher Fox, and Y. Samuel Wang

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Software for computation of maximum likelihood estimates in linear structural equation models typically employs general techniques from nonlinear optimization, such as quasi-Newton methods. In practice, careful tuning of initial values is often required to avoid convergence issues. As an alternative approach, we propose a block-coordinate descent method that cycles through the considered variables, updating only the parameters related to a given variable in each step. We show that the resulting block update problems can be solved in closed form even when the structural equation model comprises feedback cycles. Furthermore, we give a characterization of the models for which the block-coordinate descent algorithm is well defined, meaning that for generic data and starting values all block optimization problems admit a unique solution. For the characterization, we represent each model by its mixed graph (also known as path diagram), which leads to criteria that can be checked in time that is polynomial in the number of considered variables.

Article information

Ann. Statist., Volume 47, Number 2 (2019), 663-690.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation 62F10: Point estimation

Cyclic graph feedback linear structural equation model graphical model maximum likelihood estimation


Drton, Mathias; Fox, Christopher; Wang, Y. Samuel. Computation of maximum likelihood estimates in cyclic structural equation models. Ann. Statist. 47 (2019), no. 2, 663--690. doi:10.1214/17-AOS1602.

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Supplemental materials

  • Proofs of claims. The supplement provides proofs for claims made in Sections 2, 3 and 4. Specifically, we verify the form of $\det(I-B)$ as claimed in Lemma 1 and derive the likelihood equations with respect to $\Omega$ and $B$. We also verify the claims in Lemmas 4 and 5 which are required for the BCD algorithm described in the constructive proof of Theorem 1. Finally, we verify the claims in Section 4 which characterize graphs for which the BCD algorithm is well defined when initialized generically. In particular, we give a graphical condition and show that it can be checked in time which is a polynomial of the considered variables.