Let us consider two scalar stochastic processes $x_t$ and $y_t$, $t \in \mathbb{Z}$ , each observed for $T$ realizations. We assume that $x_t$ and $y_t$ are covariance stationary or that $Δx_t$ and $Δy_t$ are covariance stationary. 

If we exclude the possibility that the future can cause the past, but allow contemporaneous feedback loops due for example to temporal aggregation, there are several possibilities as regards the causal structure between $x_t$ and $y_t$ 
 - (i) The series $x_t$ has a contemporaneous or lagged causal effect on $y_t$ , i.e. $x_i → y_{i + s}$ for some $i , s$ such that $i ≥ 0, s ≥ 0$. 
 - (ii) The series $y_t$ has a contemporaneous or lagged causal effect on $x_t$ , i.e. $y_i → x_{i + s}$ for some $i , s$ such that $i ≥ 0, s ≥ 0$.
 - (iii) A not-measured series $z_t$ has a contemporaneous or lagged causal effect on both $x_t$ and $y_t$ . 
 - (iv) The causal structure between $x_t$ and $y_t$ can be described by any combination of (i)–(iii). 
 - (v) There is no causal link or path (of any type) linking $x_t$ and $y_{t + s}$ , for any $s \in \mathbb{N}$.

Cause-Effect Pairs in Time Series

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)

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 can be divided into two types:
 - Generative Models 
 - Discriminative Models

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