Judea Pearl uses the Monty Hall problem in his latest book, and when he draws out the causal network, the issue becomes obvious. The game-show host chooses whi

Judea Pearl uses the Monty Hall problem in his latest book, and when he draws out the causal network, the issue becomes obvious. The game-show host chooses which door to open based on two factors: he has to choose a door you didn't, and he has to choose a door with a goat. Hence, the door he chooses to open is probabilistically informative about your initial choice. Treating the door he opened as information on which to condition does change the posterior.