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 $$\left\{ (x_i,y_i) \right\}_{i=1}^n$$ 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 of variables $$(X, Y)$$, was introduced as a challenge at the NIPS 2008 workshop on causality. The basic scenario is: given independent and identically distributed (i.i.d.) observations $$(x_1, y_1), \dots, (x_k, y_k)$$ drawn from a distribution $$P_{X,Y}$$, infer whether $$X$$ causes $$Y$$ or $$Y$$ causes $$X$$, given the promise that exactly one is true. The NIPS 2013 challenge extended this to three classes: $$X \rightarrow Y$$, $$Y \rightarrow X$$, or a null class consisting of either the independent case $$X \perp Y$$ or the pure confounding case $$X \leftrightarrow Y$$ (shorthand for $$X \leftarrow H \rightarrow Y$$, where $$H$$ is an unobserved confounder).

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

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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