In the bivariate case, the authors assume that the variables X and Y are non Gaussian, as well as standardized to zero mean and unit variance. The goal is to distinguish between candidate linear causal models. 
The first is denoted by X → Y and defined as:
$Y:=\rho X+N_Y$ with $X \perp \!\!\! \perp N_Y$
The second model with orientation Y → X is defined as: 
$X:=\rho Y+N_X$ with $Y \perp \!\!\! \perp N_X$
The parameter ρ is the same in the two models because it is equal to the correlation coefficient between X and Y

LiNGAM Model

A theoretical identifiability has been derived by Shimizu et al. who prove that if $P_{X , Y}$ admits the linear model $Y := aX + N_Y$ with $X\perp \! \! \! \perp N_Y$  (model 1), then there exist $b\in R$ and a random variable $N_X$ such that $X := bY + N_X$ with $Y\perp \! \! \! \perp N_X$ (model 2) if and only if $X$ and $N_Y$ are Gaussian.

In different words there is only one non-identifiable case corresponding to the linear Gaussian case. Moreover if $X$ or $N_Y$ is non-Gaussian, when the candidate linear model with orientation X → Y holds, the candidate linear model with orientation Y → X does not hold. 

LiNGAM Identifiability Result

The candidate models correspond to a restrictive class of mechanism and the comparison of the candidate models is based on comparison of fit scores at fixed complexity. The fit score used for comparison is the likelihood score derived by Hyvärinen and Smith in this case as: 
$S_{X\rightarrow Y} = -[\sum_{i=1}^n G_X(x^i)+G_{N_Y}(\frac{y^i - \rho x^i}{\sqrt{1-\rho^2}})]+n\log (1-\rho^2) $
where $G_X ( u ) = \log P_X ( u )$ and $G_{N_Y}$ is the standardized log probability distribution function of the residual when regressing Y on X . $S_{Y → X}$ is computed similarly and the causal direction is decided by comparing $S_{X → Y}$ with $S_{Y → X}$.

LiNGAM Practical Evaluation

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)

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

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.

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.

Learn Before

Related

Learn After