You are all cordially invited to the AMLab seminar at Tuesday June 7 at 16:00 in C3.163, where Thijs van Ommen will give a talk titled “Robust probability updating”. Afterwards there are the usual drinks and snacks!
Abstract: In the well-known Monty Hall problem, a car is hidden behind one of three doors, and the contestant wants to compute the probabilities of where the prize is hidden given partial information (a ‘message’) from the quizmaster. Most analyses of this problem assume that the quizmaster uses a fair coin flip to decide what message to give, whenever he has a choice. We don’t make this assumption, but instead use game theory to find a strategy for the contestant that works well against any strategy the quizmaster might use. With this approach, we can also deal with a large generalization of the problem: to any finite number of doors, with any initial distribution of the winning door, and with an arbitrary set of messages (subsets of doors) from which the quizmaster can choose. In Bayesian terms, this translates to computing a posterior distribution without knowing the full joint distribution. It turns out that in general, the optimal strategies for both players in this game depend on the loss function used to evaluate the contestant’s posterior distribution. However, for certain classes of message sets, there is a single optimal posterior that does not depend on the loss function, so that we obtain an objective and general answer to how one should update probabilities in the light of new information.