Identifiability of Gaussian structural equation models with equal error variances
This work addresses the problem of causal inference in observational data for researchers in statistics and machine learning, providing a theoretical advance beyond Markov equivalence classes.
The paper proves that Gaussian structural equation models with equal error variances are fully identifiable, allowing the directed acyclic graph to be recovered from the joint Gaussian distribution, which enables causal structure inference from observational data under these conditions.
We consider structural equation models in which variables can be written as a function of their parents and noise terms, which are assumed to be jointly independent. Corresponding to each structural equation model, there is a directed acyclic graph describing the relationships between the variables. In Gaussian structural equation models with linear functions, the graph can be identified from the joint distribution only up to Markov equivalence classes, assuming faithfulness. In this work, we prove full identifiability if all noise variables have the same variances: the directed acyclic graph can be recovered from the joint Gaussian distribution. Our result has direct implications for causal inference: if the data follow a Gaussian structural equation model with equal error variances and assuming that all variables are observed, the causal structure can be inferred from observational data only. We propose a statistical method and an algorithm that exploit our theoretical findings.