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.

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 .

Practical Evaluation Based on Independence Score (ANM-HSIC)

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.

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

Many approaches basing their inference on predefined features use as input of the classifier already known models for pairwise causal discovery in a stacked classifier fashion. 

Examples of used algorithms are the Additive Noise Model (ANM) model or the Information Geometric Causal Inference (IGCI) model. These two models are the most employed, as they represent a decent tradeoff between performance and computational cost. 

Preexisting Pairwise Causal Discovery Algorithms

 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 (ANM)

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)

Information geometric causal inference takes on another approach to infer the causal direction in the pairwise setting. It bases itself on the independence between the cause and the causal mechanism: under the strong assumption that X and Y are related by a bijective relation, the cause $P_X$ is independent from the mechanism $P_{Y | X}$ and not in the opposite direction. This approach can also be related to a complexity approach on the mechanisms

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