Zhang and Hyvärinen proved that the PNL model is generally identifiable, saying that if $P_{X , Y}$ satisfies a PNL model X → Y , $P_{X , Y}$ cannot satisfy a PNL Y → X , except when specific conditions are encountered. The set of non-identifiable distribution $P_{X , Y}$ is larger than for ANM. However PNL is more general and can handle more types of observed distribution.

PNL Identifiability Result

The model has been evaluated in using an independence score between cause and noise. 
As $\hat{g}_Y$ is assumed to be invertible, the idea is that the noise variable can be recovered from $P_{X , Y}$ as: 
$\hat{N}_Y = g_Y^{-1}(Y) - f_Y(X)$
The noise variable is then estimated by functions $l_1$ and $l_2$ such as $\hat{Y}_Y = l_1(Y) - l_2(X)$ with $\hat{N}_Y$ independent of X. It comes back to solve a constrained nonlinear ICA problem, that can be achieved by minimizing $I(X,\hat{N}_Y; \theta)$, the mutual information between X and $\hat{N}_Y$ with respect to the parameter of the model θ . Symmetrically, an optimization of $I(X,\hat{N}_Y; \theta)$ is performed. 
The causal direction X → Y is preferred if $I(X,\hat{N}_Y; \hat{\theta})< I(Y,\hat{N}_X; \hat{\theta)}$, Y → X otherwise.

PNL Practical Evaluation

Post-NonLinear model (PNL) is a generalization of ANM called that takes into account nonlinear interactions between the cause and the noise has been proposed by Zhang and Hyvärinen.

A bivariate Post-NonLinear Model (PNL) $X → Y$ is defined as: 
$Y:= \hat{g}_Y (\hat{f}_Y(X)+N_Y)$ with $X \perp \! \! \! \perp N_Y$
where $\hat{f}_Y: \mathbb{R} \rightarrow \mathbb{R}$ and $\hat{g}_Y : \mathbb{R} \rightarrow \mathbb{R}$ are two Borel measurable functions. $\hat{g}_Y$ is assumed to be invertible.

University of Michigan - Ann Arbor

This kind of methods range from very simple class of functions with LINGAM to more complex class of functions such as PNL, with each time a trade-off between the identifiability and the generality of the proposed approach as shown in the picture.

Indeed, when the class of function is very restricted, there are fewer non identifiable cases, but the model can only successfully be used when encountering very specific observed data (such as data generated by linear mechanisms). When the class of functions becomes larger, the model becomes more general and can be used for more types of distribution, but at the cost of more non identifiable cases. In the extreme case of a completely general model without restriction on the class of mechanisms, all pairs become non identifiable.

Examples and Reviews of Methods with a Restricted Class of Mechanisms

 Isabelle Guyon, Alexander Statnikov, Berna Bakir Batu (2019)
Cause Effect Pairs in Machine Learning (The Springer Series on Challenges in Machine Learning)
https://www.amazon.com/Effect-Machine-Learning-Springer-Challenges/dp/3030218090

Cause Effect Pairs in Machine Learning

The LiNGAM method was first developed for directed acyclic graph orientation for more than two variables. LiNGAM handles linear structural equation models, where each variable is continuous and modeled as:
$X_i := \sum_{k} \alpha_k P_a^k (X_i) + E_i, i \in [[1,n]]$
with the k th parent of $X_i$ and $α_k$ a real value. Assuming further that all probability distributions of source nodes in the causal graph are non-Gaussian, it is shown that the causal structure is fully identifiable (all edges can be oriented)

Pairwise LiNGAM Inference

Additive Noise Model is an extension of LiNGAM model to deal with nonlinear mechanism derived by Hoyer et al. 
A bivariate additive noise model (ANM) X → Y is defined as: 
$Y:=f_Y(X) + N_Y$ with $X \perp \! \! \! \perp N_Y$
where $f_Y : \mathbb{R}\rightarrow \mathbb{R}$ is a Borel measurable function.

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