In the bivariate case of the Linear Non-Gaussian Acyclic Model (LiNGAM), the variables $$X$$ and $$Y$$ are assumed to be non-Gaussian, and standardized to zero mean and unit variance. The goal is to distinguish between candidate linear causal models. The first model, denoted by $$X \rightarrow Y$$, is defined as: $$Y := \rho X + N_Y$$ with $$X \perp \!\!\! \perp N_Y$$. The second model, with orientation $$Y \rightarrow X$$, is defined as: $$X := \rho Y + N_X$$ with $$Y \perp \!\!\! \perp N_X$$. The parameter $$\rho$$ is the same in both models because it is equal to the correlation coefficient between $$X$$ and $$Y$$.

Bivariate 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 $$b \in \mathbb{R}$$ 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 other 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 $$X \rightarrow Y$$ holds, the candidate linear model with orientation $$Y \rightarrow X$$ does not hold.

LiNGAM Identifiability Result

The Linear Non-Gaussian Acyclic Model (LiNGAM) was originally developed for directed acyclic graph orientation involving more than two variables. LiNGAM handles linear structural equation models, where each continuous variable is 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 $$\alpha_k$$ as 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 uniquely oriented).

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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

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)

Linear Non-Gaussian Acyclic Model (LiNGAM)

The Post-NonLinear (PNL) model is a generalization of the Additive Noise Model (ANM) that takes into account nonlinear interactions between the cause and the noise, as proposed by Zhang and Hyvärinen. A bivariate Post-NonLinear Model (PNL) $$X \rightarrow Y$$ 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 Borel measurable functions, and $$\hat{g}_Y$$ is assumed to be invertible.

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