Hoyer et al. proved that the ANM model is generally identifiable, saying that if $P_{X , Y}$ satisfies an additive noise model with orientation X → Y , $P_{X , Y}$ cannot satisfy an additive model with orientation Y → X . 
However, specific cases have been identified where ANM is non-identifiable. In particular the linear Gaussian case is non identifiable.

University of Michigan - Ann Arbor

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.

Additive Noise Model (ANM)

 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

ANM Identifiability Result

 In the original paper, the ANM is seen as a discriminative model and the fit score is based on an independence test between noise and cause. More precisely for the two alternatives X → Y and Y → X , the estimated mechanisms and are obtained via Gaussian process regressions. These estimated regression functions are used to estimate the residuals $\hat{n}_Y = y-\hat{f}_Y(x)$ and $\hat{n}_X = x - \hat{f}_X(y)$. The scores $S_{X → Y}$ and $S_{Y → X}$ correspond respectively to kernel HSIC independence test between $\hat{n}_Y$ and x (for X → Y ) and between $\hat{n}_X$ and y (for Y → X ). 

If $S_{X → Y} < S_{Y → X}$ it is decided that X → Y . If $S_{X → Y} > S_{Y → X}$ it is decided that Y → X .

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