The Book of Why: The New Science of Cause and Effect (Penguin Science)

by Judea Pearl

A → B ← C. This is the most fascinating junction, called a “collider.” Felix Elwert and Chris Winship have illustrated this junction using three features of Hollywood actors: Talent → Celebrity ← Beauty. Here we are asserting that both talent and beauty contribute to an actor’s success, but beauty and talent are completely unrelated to one another in the general population. We will now see that this collider pattern works in exactly the opposite way from chains or forks when we condition on the variable in the middle. If A and C are independent to begin with, conditioning on B will make them dependent. For example, if we look only at famous actors (in other words, we observe the variable Celebrity = 1), we will see a negative correlation between talent and beauty: finding out that a celebrity is unattractive increases our belief that he or she is talented. This negative correlation is sometimes called collider bias or the “explain-away” effect. For simplicity, suppose that you don’t need both talent and beauty to be a celebrity; one is sufficient. Then if Celebrity A is a particularly good actor, that “explains away” his success, and he doesn’t need to be any more beautiful than the average person. On the other hand, if Celebrity B is a really bad actor, then the only way to explain his success is his good looks. So, given the outcome Celebrity = 1, talent and beauty are inversely related—even though they are not related in the population as a whole.