Some General Identification Results for Linear Latent Hierarchical Causal Structure

Abstract

We study the problem of learning hierarchical causal structure among latent variables from measured variables. While some existing methods are able to recover the latent hierarchical causal structure, they mostly suffer from restricted assumptions, including the tree-structured graph constraint, no ``triangle" structure, and non-Gaussian assumptions. In this paper, we relax these restrictions above and consider a more general and challenging scenario where the beyond tree-structured graph, the ``triangle" structure, and the arbitrary noise distribution are allowed. We investigate the identifiability of the latent hierarchical causal structure and show that by using second-order statistics, the latent hierarchical structure can be identified up to the Markov equivalence classes over latent variables. Moreover, some directions in the Markov equivalence classes of latent variables can be further identified using partially non-Gaussian data. Based on the theoretical results above, we design an effective algorithm for learning the latent hierarchical causal structure. The experimental results on synthetic data verify the effectiveness of the proposed method.