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

by Judea Pearl

A Bayesian network is literally nothing more than a compact representation of a huge probability table. The arrows mean only that the probabilities of child nodes are related to the values of parent nodes by a certain formula (the conditional probability tables) and that this relation is sufficient. That is, knowing additional ancestors of the child will not change the formula. Likewise, a missing arrow between any two nodes means that they are independent, once we know the values of their parents. We saw a simple version of this statement earlier, when we discussed the screening-off effect in chains and links. In a chain A → B → C, the missing arrow between A and C means that A and C are independent once we know the values of their parents. Because A has no parents, and the only parent of C is B, it follows that A and C are independent once we know the value of B, which agrees with what we said before.