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.

University of Michigan - Ann Arbor

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.

Post Nonlinear Model (PNL)

 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

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.

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