The Monty Hall Problem (2 of 3): What is the Optimal Policy?
If you have experimented with the network in BayesiaLab or with the WebSimulator, you have probably seen that the optimal policy is to switch door.
Indeed, as illustrated in the simulator below, if you set Your Door = Door #1, then Monty cannot open Door #1 (grayed out in the Door Opened component). If you then set Door Opened = Door #2, you will see that the probability that the car is behind Door #1 is still 1/3, while the probability that the car is behind Door #3 has increased to 2/3. The optimal decision is then to switch from Door #1 to Door #2. Notice that this is the optimal decision when we do not take into account any additional dimension. The situation might be different if we were to add the psychological cost of discovering, after switching, that your initial choice was the right one.