Information Geometric Causal Inference (IGCI) is an approach used to infer causal direction in a pairwise setting. It is based on the independence between the cause and the causal mechanism: under the strong assumption that variables $$X$$ and $$Y$$ are related by a bijective relation, the distribution of the cause $$P_X$$ is independent from the mechanism $$P_{Y | X}$$, while this independence does not hold in the opposite direction. This approach is also related to complexity-based methods on causal mechanisms.

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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

Methods exploiting the independence between cause and mechanism infer causal direction based on the principle that when $$X \to Y$$, the distribution of the cause $$P_X$$ should not contain information that can be useful to derive the conditional model $$Q_{Y | X}$$ on the data. This approach leverages the notion of complexity of causal mechanisms.

Methods Exploiting the Independence between Cause and Mechanism

 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)

Information Geometric Causal Inference (IGCI)

Here are some examples of the methods that exploit the independence between cause and mechanism:
 - NonLinear Deterministic Mechanism
 - Unsupervised Inverse Regression
 - Causal Inference via Kernel Deviance Measures
 - Cause-Effect Inference by Comparing Regression Errors

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