On Estimation of Functional Causal Models: General Results and Application to the Post-Nonlinear Causal Model

Kun Zhang, Zhikun Wang, Jiji Zhang, Bernhard Schölkopf

Research output: Contribution to journalJournal articlepeer-review

40 Citations (Scopus)


Compared to constraint-based causal discovery, causal discovery based on functional causal models is able to identify the whole causal model under appropriate assumptions [Shimizu et al. 2006; Hoyer et al. 2009; Zhang and Hyvärinen 2009b]. Functional causal models represent the effect as a function of the direct causes together with an independent noise term. Examples include the linear non-Gaussian acyclic model (LiNGAM), nonlinear additive noise model, and post-nonlinear (PNL) model. Currently, there are two ways to estimate the parameters in the models: dependence minimization and maximum likelihood. In this article, we show that for any acyclic functional causal model, minimizing the mutual information between the hypothetical cause and the noise term is equivalent to maximizing the data likelihood with a flexible model for the distribution of the noise term. We then focus on estimation of the PNL causal model and propose to estimate it with the warped Gaussian process with the noise modeled by the mixture of Gaussians. As a Bayesian nonparametric approach, it outperforms the previous one based on mutual information minimization with nonlinear functions represented by multilayer perceptrons; we also show that unlike the ordinary regression, estimation results of the PNL causal model are sensitive to the assumption on the noise distribution. Experimental results on both synthetic and real data support our theoretical claims.
Original languageEnglish
Article number13
Pages (from-to)1–22
Number of pages22
JournalACM Transactions on Intelligent Systems and Technology
Issue number2
Publication statusPublished - Dec 2015

User-Defined Keywords

  • Systems and Information Theory
  • Learning
  • Probability and Statistics
  • Causal discovery
  • functional causal model
  • post-nonlinear causal model
  • statistical independence
  • maximum likelihood


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