21.2 The Formal Notation of Causality

A common mistake is defining causation using probability:

$$X \text{ causes } Y \text{ if } P(Y | X) > P(Y).$$

Seeing $X$ (1st level) doesn’t mean the probability of Y increases.

It could be either that

1. $X$ causes Y, or
2. $Z$ affects both $X$ and $Y$. We might be able use control variables - $P(Y|X, Z = z) > P(Y|Z = z)$. But then the question becomes
   1. How to choose $Z$?
   2. Did you choose enough $Z$?
   3. Did you choose the right $Z$?

Hence, the previous statement is incorrect. The correct causal statement is:

$$P(Y | do(X)) > P(Y).$$

With causal diagrams and do-calculus, we can formally express interventions and answer questions at the 2nd level (Intervention).