The Book of Why: The New Science of Cause and Effect

by Judea Pearl

Now let me discuss the practical objection to Bayes’s rule—which may be even more consequential when we exit the realm of theology and enter the realm of science. If we try to apply the rule to the billiard-ball puzzle, in order to find P(L | x) we need a quantity that is not available to us from the physics of billiard balls: we need the prior probability of the length L, which is every bit as tough to estimate as our desired P(L | x). Moreover, this probability will vary significantly from person to person, depending on a given individual’s previous experience with tables of different lengths. A person who has never in his life seen a snooker table would be very doubtful that L could be longer than ten feet. A person who has only seen snooker tables and never seen a billiard table would, on the other hand, give a very low prior probability to L being less than ten feet. This variability, also known as “subjectivity,” is sometimes seen as a deficiency of Bayesian inference. Others regard it as a powerful advantage; it permits us to express our personal experience mathematically and combine it with data in a principled and transparent way. Bayes’s rule informs our reasoning in cases where ordinary intuition fails us or where emotion might lead us astray. We will demonstrate this power in a situation familiar to all of us.