Discriiminative algorithms do not attempt to reverse engineer the data generating process, they merely classify the empirical joint distribution of two variables $X$ and $Y$ (a scatter plot) as being a $X$ cause $Y$ or a $Y$ cause $X$ (or neither).

Discriminative Models for Cause-Effect Problem

Generative algorithms solve the cause-effect pair problem from a reverse engineering perspective: given the available measurements $\{ (x_i,y_i) \}_{i=1}^n $ of the two variables $X$ and $Y$, the task is to discover the underlying causal process that led the variables to take the values they have.

Such methods allow gaining not only a clue about the causal direction but also information about the mechanism itself, making causal discovery less of a black box decision process.

Generative Models for Cause-Effect Problem

Cause-effect methods can be divided into two types:
 - Generative Models 
 - Discriminative Models

University of Michigan - Ann Arbor

Telling cause from effect from purely observational data (focusing on pairs (X,Y) of variables rather than multi-variate case) has been a challenge at the NIPS 2008 workshop on Causality.  The scenario can be described as follows: 
 - Given observations $( x_1 , y_1 ), …, ( x_k , y_k )$ iid (independent and identically distributed) drawn from some distribution $P_{X , Y}$ , infer whether X causes Y (a) or Y causes X (b) , given the promise that exactly one of these alternatives is true. 

The NIPS 2013 challenge extended this setting to three classes:
$X\rightarrow Y (a)$ , $Y\rightarrow X (b)$, or the null class $\{(c),(d)\}$, which consisted of the unconnected case $X Y$ or the case of pure confounding $X \leftrightarrow Y$ (this notation is used as shorthand for X ← H → Y , where H is unobserved, but where neither X or Y cause each other). 

Cause-Effect Pair Challenge

 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

Why considering only pairs of variables?

There are multiple justifications to embrace the cause-effect pair setting. 
 - Practical (the primary justification)
 - Fundamental Scientific Interest
 - Low Dimensionality

However, what is gained on one side might be lost on another and ultimately, so it is better to think of pairwise methods as complements to multi-variate graphical model methods that help orienting edges left un-oriented in Markov equivalence classes.

Justifications to Embrace the Cause-Effect Pair Setting

Cause-Effect Methods

There are two important extensions of the cause-effect pair problem:
 - Causal discovery methods for time series
 - Going beyond the treatment of two variables (triplet and more)

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