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What is the mathematical definition of a causal relationship between two random variables?

Given a sample from the joint distribution of two random variables $X$ and $Y$, when would we say $X$ causes $Y$?

For context, I am reading this paper about causal discovery.

What is the mathematical definition of a causal relationship between
two random variables?

Mathematically, a causal model consists of functional relationships between variables. For instance, consider the system of structural equations below:

$$
x = f_x(epsilon_{x})\
y = f_y(x, epsilon_{y})
$$

This means that $x$ functionally determines the value of $y$ (if you intervene on $x$ this changes the values of $y$) but not the other way around. Graphically, this is usually represented by $x rightarrow y$, which means that $x$ enters the structural equation of y. As an addendum, you can also express a causal model in terms of joint distributions of counterfactual variables, which is mathematically equivalent to functional models.

Given a sample from the joint distribution of two random variables X
and Y, when would we say X causes Y?

Sometimes (or most of the times) you do not have knowledge about the shape of the structural equations $f_{x}$, $f_y$, nor even whether $xrightarrow y$ or $y rightarrow x$. The only information you have is the joint probability distribution $p(y,x)$ (or samples from this distribution).

This leads to your question: when can I recover the direction of causality just from the data? Or, more precisely, when can I recover whether $x$ enters the structural equation of $y$ or vice-versa, just from the data?

Of course, without any fundamentally untestable assumptions about the causal model, this is impossible. The problem is that several different causal models can entail the same joint probability distribution of observed variables. The most common example is a causal linear system with gaussian noise.

But under some causal assumptions, this might be possible---and this is what the causal discovery literature works on. If you have no prior exposure to this topic, you might want to start from Elements of Causal Inference by Peters, Janzing and Scholkopf, as well as chapter 2 from Causality by Judea Pearl. We have a topic here on CV for references on causal discovery, but we don't have that many references listed there yet.

Therefore, there isn't just one answer to your question, since it depends on the assumptions one makes. The paper you mention cites some examples, such as assuming a linear model with non-gaussian noise. This case is known as LINGAN (short for linear non-gaussian acyclic model), here is an example in R:

library(pcalg)

set.seed(1234)

n <- 500

eps1 <- sign(rnorm(n)) * sqrt(abs(rnorm(n)))

eps2 <- runif(n) - 0.5

x2 <- 3 + eps2

x1 <- 0.9*x2 + 7 + eps1



# runs lingam

X <- cbind(x1, x2)

res <- lingam(X)

as(res, "amat") 



# Adjacency Matrix 'amat' (2 x 2) of type ‘pag’:

#     [,1]  [,2]

# [1,] .     .   

# [2,]  TRUE .     

Notice here we have a linear causal model with non-gaussian noise where $x_2$ causes $x_1$ and lingam correctly recovers the causal direction. However, notice this depends critically on the LINGAM assumptions.

For the case of the paper you cite, they make this specific assumption (see their "postulate"):

If $xrightarrow y$ , the minimal description length of the mechanism mapping X to Y is independent of the value of X, whereas the minimal description length of the mechanism mapping Y to X is dependent on the value of Y.

Note this is an assumption. This is what we would call their "identification condition". Essentially, the postulate imposes restrictions on the joint distribution $p(x,y)$. That is, the postulate says that if $x rightarrow y$ certain restrictions holds in the data, and if $y rightarrow x$ other restrictions hold. These types of restrictions that have testable implications (impose constraints on $p(y,x)$) is what allows one to recover directionally from observational data.

As a final remark, causal discovery results are still very limited, and depend on strong assumptions, be careful when applying these on real world context.

There are a variety of approaches to formalizing causality (which is in keeping with substantial philosophical disagreement about causality that has been around for centuries). A popular one is in terms of potential outcomes. The potential-outcomes approach, called the Rubin causal model, supposes that for each causal state of affairs, there's a different random variable. So, $Y_1$ might be the random variable of possible outcomes from a clinical trial if a subject takes the study drug, and $Y_2$ might be the random variable if he takes the placebo. The causal effect is the difference between $Y_1$ and $Y_2$. If in fact $Y_1 = Y_2$, we could say that the treatment has no effect. Otherwise, we could say that the treatment condition causes the outcome.

Causal relationships between variables can also be represented with directional acylical graphs, which have a very different flavor but turn out to be mathematically equivalent to the Rubin model (Wasserman, 2004, section 17.8).

Wasserman, L. (2004). All of statistics: A concise course in statistical inference. New York, NY: Springer. ISBN 978-0-387-40272-7.

There are two ways to determine whether $X$ is the cause of $Y$. The first is standard while the second is my own claim.

1. There exists an intervention on $X$ such that the value of $Y$ is changed

An intervention is a surgical change to a variable that does not affect variables it depends on. Interventions have been formalized rigorously in structural equations and causal graphical models, but as far as I know, there is no definition which is independent of a particular model class.

2. The simulation of $Y$ requires the simulation of $X$
   

To make this rigorous requires formalizing a model over $X$ and $Y$, and in particular the semantics which define how it is simulated.

In modern approaches to causation, intervention is taken as the primitive object which defines causal relationships (definition 1). In my opinion, however, intervention is a reflection of, and necessarily consistent with simulation dynamics.