Identification of Nonlinear Latent Hierarchical Causal Models

Counterfactual reasoning entails the identification of causal models. However, the task of identifying latent variables and causal structures from observational data can be highly challenging, especially when observed variables are generated by causally related latent variables with nonlinear functions. In this work, we investigate the identification problem for nonlinear latent hierarchical models in which observed variables are generated by causally related latent variables, and some latent variables may not have observed children. We show that the identifiability of both causal structure and latent variables can be achieved under mild assumptions: on causal structures, we allow for multiple paths between any pair of variables, which relaxes latent tree assumptions in prior work; on structural functions, we do not make parametric assumptions, thus permitting general nonlinearity and multi-dimensional continuous variables. Specifically, we first develop a basic identification criterion in the form of novel identifiability guarantees for an elementary latent variable model. Leveraging this criterion, we show that both causal structures and latent variables of the hierarchical model can be identified asymptotically by explicitly constructing an estimation procedure. To the best of our knowledge, our work is the first to establish identifiability guarantees for both causal structures and latent variables in nonlinear latent hierarchical models.