Consistent Bayesian causal discovery for structural equation models with equal error variances

We consider the problem of recovering the true causal structure among a set of variables, generated by a linear acyclic structural equation model (SEM) with the error terms being independent, not necessarily Gaussian, and having equal variances. It is well-known that the true underlying directed acyclic graph (DAG) encoding the causal structure is uniquely identifiable under this assumption. Interestingly, in this setting, it further holds that the sum of minimum expected squared errors for every variable, while predicted by the best linear combination of its parent variables, is minimised if and only if the causal structure is represented by any supergraph of the true DAG. In this work, we propose a Bayesian DAG selection method, where the working model assumes Gaussian SEM with equal error variances, and employ independent g-priors on each set of SEM coefficients. Furthermore, we utilise the aforementioned key property to establish that the proposed method recovers the true graph consistently without any additional distributional assumption, and illustrate it with a simulation study.