Identifiability of Gaussian Structural Equation Models with Same Error Variances

We consider structural equation models (SEMs) in which variables can be written as a function of their parents and noise terms (the latter are assumed to be jointly independent). Corresponding to each SEM, there is a directed acyclic graph (DAG) G_0 describing the relationships between the variables. In Gaussian SEMs with linear functions, the graph can be identified from the joint distribution only up to Markov equivalence classes (assuming faithfulness). It has been shown, however, that this constitutes an exceptional case. In the case of linear functions and non-Gaussian noise, the DAG becomes identifiable. Apart from few exceptions the same is true for non-linear functions and arbitrarily distributed additive noise. In this work, we prove identifiability for a third modification: if we require all noise variables to have the same variances, again, the DAG can be recovered from the joint Gaussian distribution. Our result can be applied to the problem of causal inference. If the data follow a Gaussian SEM with same error variances and given that all variables are observed, the causal structure can be inferred from observational data only.