On the Identifiability of the Post-Nonlinear Causal Model

TL;DR

This paper investigates the identifiability of the post-nonlinear causal model, establishing conditions for when it can reliably distinguish cause from effect in two-variable cases and extending the approach to multiple variables.

Contribution

It provides a systematic analysis of the model's identifiability, offering sufficient conditions and a practical method for causal structure discovery with more than two variables.

Findings

Model is identifiable in most cases for two variables

Sufficient conditions for identifiability are provided

Method extends to multiple variables avoiding exhaustive search

Abstract

By taking into account the nonlinear effect of the cause, the inner noise effect, and the measurement distortion effect in the observed variables, the post-nonlinear (PNL) causal model has demonstrated its excellent performance in distinguishing the cause from effect. However, its identifiability has not been properly addressed, and how to apply it in the case of more than two variables is also a problem. In this paper, we conduct a systematic investigation on its identifiability in the two-variable case. We show that this model is identifiable in most cases; by enumerating all possible situations in which the model is not identifiable, we provide sufficient conditions for its identifiability. Simulations are given to support the theoretical results. Moreover, in the case of more than two variables, we show that the whole causal structure can be found by applying the PNL causal model to each structure in the Markov equivalent class and testing if the disturbance is independent of the direct causes for each variable. In this way the exhaustive search over all possible causal structures is avoided.