Consider a pair of random variables $( X , Y )$ with ground truth $G ∈\{ X → Y , X ← Y , X \leftrightarrow Y , X ⊥ Y \} $, drawn from an unknown mother distribution. We call an estimator $\hat{G}$ of $G$ based on $( X , Y )$. 

The authors define a causation coefficient as follows:
A causation coefficient $C ( X , Y )$ is a real scalar value, such that the larger $C ( X , Y )$, the more confident we are that $G = X → Y$ . A causation coefficient must have the following desired properties:
1. [Anti-symmetry] 
$C(X,Y) = -C(Y,X)$
2. [Discriminant] 
$C(X,Y)>\theta \leq 0 \Rightarrow \hat{G} = [X\rightarrow Y]$ (and $\hat{G} \neq [X\rightarrow Y]$ otherwise)
3. [Arbitrary units] 
$C(aX+b, cY+d) = C(X,Y); a,b,c,d\in R, a\neq 0, c\neq 0$

Causation Coefficient

While talking about methods of evaluation of cause-effect pair algorithms, there are three points of view : 
 - that of algorithm developers in need to characterize their new algorithms both theoretically and empirically
 - that of practitioners who want to ensure the validity of a potential discovery
 - that of benchmark organizers who want to make fair comparisons.

Evaluation Methods for Cause-Effect Pair Algorithms

For practical reasons, it is convenient to recast the problem as a two-class classification problem separating X → Y from all other cases. 

We need to build a classifier (eventually using training data) to predict the ground truth of causal relationships as accurately as possible. The authors propose to construct a causation coefficient, which is a discriminant function in this two-class classification problem.

University of Michigan - Ann Arbor

The authors can cast the cause-effect pair problem as a regular machine learning classification problem in which input-output pairs $( P_{Π = π} ( X , Y )$, G = g ) are drawn randomly and independently from a $\textbf{“mother distribution” }$ $P_M( P_{Π = π} ( X , Y ), G = g )$, then divided into training and test sets. 

The problem is to “classify” distributions $P_{Π = π} ( X , Y )$ into one of four classes $g ∈ \{ X → Y , X ← Y , X \leftrightarrow Y , X ⊥ Y \} $

Cast Cause-Effect Pair Problem as Machine Learning Classification Problem

 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 mother distribution is a distribution over join distributions of pairs of variables. The authors adopt the following definition of a mother distribution:

A mother distribution $P_M( P_{Π = π} ( X , Y ), G = g )$ is a distribution over pairs $( P_{Π = π} ( X , Y ), G = g )$, where $Π$ and $G$ are random variables:
 - $Π = π$ indexes distribution $P_{Π = π} ( X , Y )$ over random variables $X$ and $Y$ ; $π$ may possibly take an infinite number of values (e.g. parameters of a generative process)
 - $G = g$ labels causal relationship, $g ∈ \{ X → Y , X ← Y , X \leftrightarrow Y , X ⊥ Y \}$.

Mother Distribution

Recast as a Two-Class Classification Problem

The authors consider that pairs of variables X and Y can be in one of four types of relationships: 
 - X causes Y ( X → Y )
 - Y causes X ( X ← Y )
 - X and Y are dependent, but not in a causal relationship ( X :left_right_arrow: Y ) 
 - X and Y are independent ( X ⊥ Y ). 
Any pair of variables ( X , Y ) is associated with one and only one such relationships, called “ground truth” G . By definition, we have:
$G=[X\rightarrow Y] \Rightarrow \exists f \wedge \exists N_y s.t. Y:=f(X,N_y)$
$G=[X\leftarrow Y] \Rightarrow \exists f \wedge \exists N_x s.t. X:=f(Y,N_x)$
$G=[X\leftrightarrow Y] \Rightarrow \exists f,g \wedge \exists N_x, N_y, Z s.t. X:= f(Z,N_x)\wedge y:= f(Z,N_y)$
$G= X ⊥ Y$  X and Y are independent, no functional relationship

The authors do not necessarily assume that ( X ⊥ N y ) nor that ( Y ⊥ N x ), thus there can be additional latent confounders in the first two cases X → Y and X ← Y.

Definition of Pairwise Causal and Non-causal Relationship (in the book)

A distinctive characteristic of the causal pairwise classification problem compared to the traditional classification problem is the nature of the samples points. In a regular classification problem a sample is a vector representing the position of the example in the feature space $\mathbb{R}^d$, where $d$ represents the number of features. In our pairwise classification problem, a sample is an empirical distribution, a set of points $\{  x_i,y_i \}_{i=1}^{n_j}$. 
Therefore, a feature construction step is added between the data and the learning algorithm.

General Structure of Discriminant Learning Algorithms for Pairwise Causal Discovery

There are two primary limitations of the Mother Distribution framework where we cast the pairwise causal discovery problem as a supervised learning problem, regarding the quality and the quantity of the training data respectively.

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