In Pairwise LiNGAM inference, the candidate linear causal models are evaluated by comparing their fit scores at a fixed complexity. The fit score used for comparison is the likelihood score, derived by Hyvärinen and Smith as:

$$S_{X \rightarrow Y} = -\left[\sum_{i=1}^n G_X(x^i)+G_{N_Y}\left(\frac{y^i - \rho x^i}{\sqrt{1-\rho^2}}\right)\right]+n\log (1-\rho^2)$$

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 $$X$$. The score $$S_{Y \rightarrow X}$$ is computed similarly, and the causal direction is decided by comparing $$S_{X \rightarrow Y}$$ with $$S_{Y \rightarrow X}$$.

LiNGAM Practical Evaluation

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 $$Y$$.

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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).

Linear Non-Gaussian Acyclic Model (LiNGAM)

 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

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.

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