On the Identifiability of the Post-Nonlinear Causal Model

By Kun Zhang, Aapo Hyvärinen. 2009

644 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Control and Systems Engineering, Probabilistic & Bayesian Methods, Signal Processing

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Citations in the corpus

Gradient-Based Neural DAG Learning (2019, background): “There are many examples of graph identifiability results for continuous variables [11, 9, 14, 17] as well as for discrete variables [10].”
Toward Causal Representation Learning (2021, methodology): “It generalizes results of Shimizu et al. [229] for linear functions, and it can be generalized to include non-linear rescalings [279], loops [174], confounders [123], and multivariable settings [189].”
Sample-Efficient Reinforcement Learning via Counterfactual-Based Data Augmentation (2020, background): “, linear relations [13], nonlinear relations with additive noise [14], or the causal model with further post-nonlinear transformations [15].”
Algorithmic recourse under imperfect causal knowledge: a probabilistic approach (2020, background): “Of particular interest for our setting is the post-nonlinear model of [51], which refers to the setting in which a non-linearity g is applied on top of an ANM , i.e., X r := g r ( f r ( X pa ( r ) ) + U r ) , and for which complete conditions on { f r , g r } have been provided that lead to…”
Causality for Machine Learning (2019, background): “It generalizes results of Shimizu et al. (2006) for linear functions, and it can be generalized to include nonlinear rescalings (Zhang and Hyvärinen, 2009), loops (Mooij et al., 2011), confounders (Janzing et al., 2009), and multi-variable settings (Peters et al., 2011).”
Causal Discovery from Heterogeneous/Nonstationary Data (2019, methodology): “This restricts the causal direction identiﬁcation to certain functional classes, e.g., additive (Shimizu et al., 2006; Hoyer et al., 2009; Zhang and Hyv¨arinen, 2009a) and post-nonlinear models (Zhang and Chan, 2006; Zhang and Hyv¨arinen, 2009b).”
Detecting non-causal artifacts in multivariate linear regression models (2018, background): “…assumption Spirtes et al. (1993); Pearl (2000), researchers have also stated assumptions that admit causal inference when no conditional statistical inde-pendences hold, e.g., Kano and Shimizu (2003); Sun et al. (2006); Hoyer et al. (2009); Zhang and Hyvärinen (2009); Blöbaum et al. (2017).”
Causal Generative Neural Networks (2017, background): “In observational causal discovery, some authors exploit distributional asymmetries to discover bivariate causal relations [Hoyer et al. , 2009; Zhang and Hyv¨arinen, 2009; Daniusis et al. , 2012; Stegle et al. , 2010; Lopez-Paz et al. , 2015; Fonollosa, 2016], while others rely on conditional…”
Causal Discovery from Nonstationary/Heterogeneous Data: Skeleton Estimation and Orientation Determination (2017, background): “…one ﬁts a ﬁxed functional causal model (e.g., the linear, non-Gaussian model [Shimizu et al. , 2006], the additive noise model [Hoyer et al. , 2009; Zhang and Hyv¨arinen, 2009a], or the post-nonlinear causal model [Zhang and Chan, 2006; Zhang and Hyv¨arinen, 2009b]) to distribution-shifted data,…”
Consistency of Causal Inference under the Additive Noise Model (2013, background): “…et al. (2009), Zhang & Hyvärinen (2009) and Peters et al. (2011a) showed that the CAM is identifiable more generally even if f is nonlinear, the main technical requirements being that the marginals PX , and Pη are absolutely continuous on R, with Pη having support R. Note that Zhang &…”
On causal and anticausal learning (2012, background): “A similar statement has been shown for the postnonlinear ANM model (Zhang & Hyv¨arinen, 2009) , where ψ is an invertible function.”
Identifiability of Causal Graphs using Functional Models (2011, methodology): “Another example of a bivariate identifiable model class are postnonlinear models [Zhang and Hyvärinen, 2009].”
Kernel-based Conditional Independence Test and Application in Causal Discovery (2011, methodology): “We generated X and Y from Z 1 accroding to the post-nonlinear data generating procedure (Zhang and Hyv¨arinen, 2009): they were constructed as G ( F ( Z 1 ) + E ) where G and F are random mixtures of linear, cubic, and tanh functions and are diﬀerent for X and Y , and E is independent across X and…”
Inferring deterministic causal relations (2010, methodology): “, 2009], using Gaussian Process Regression which assumes Gaussian noise, and an implementation of the Post-NonLinear (PNL) model [Zhang and Hyvärinen, 2009], using neural networks for constrained nonlinear independent component analysis.”
Distinguishing Cause from Effect Using Observational Data: Methods and Benchmarks (2014, background): “Zhang and Hyvärinen (2009) proved that non-identifiable ANMs necessarily fall into one out of five classes.”
CAM: Causal Additive Models, high-dimensional order search and penalized regression (2013, background): “Closely related examples include nonlinearly transformed additive structural equation models [46] or Gaussian structural equation models with same error variances [25].”