Causal discovery from a set of observations is one of the fundamental problems across several disciplines. For continuous variables, recently a number of causal discovery methods have demonstrated their effectiveness in distinguishing the cause from effect by exploring certain properties of the conditional distribution, but causal discovery on categorical data still remains to be a challenging problem, because it is generally not easy to find a compact description of the causal mechanism for the true causal direction. In this paper we make an attempt to find a way to solve this problem by assuming a two-stage causal process: the first stage maps the cause to a hidden variable of a lower cardinality, and the second stage generates the effect from the hidden representation. In this way, the causal mechanism admits a simple yet compact representation. We show that under this model, the causal direction is identifiable under some weak conditions on the true causal mechanism. We also provide an effective solution to recover the above hidden compact representation within the likelihood framework. Empirical studies verify the effectiveness of the proposed approach on both synthetic and real-world data.
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