The maximum likelihood threshold of a path diagram

Mathias Drton, Christopher Fox, Andreas Käufl, Guillaume Pouliot

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)1536-1553
Number of pages18
JournalAnnals of Statistics
Volume47
Issue number3
DOIs
StatePublished - Jan 2019
Externally publishedYes

Keywords

  • Covariance matrix
  • Graphical model
  • Maximum likelihood
  • Normal distribution
  • Path diagram
  • Structural equation model

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