How should we update a probability distribution when given new information? The standard answer is: "conditioning", but this is not always correct: as shown by examples such as the three prisoners problem, the Monty Hall (three cars, quizmaster) problem and the two-children puzzle, at least when applied in the common "naive" form, conditioning often gives the wrong answers. We give a detailed explanation of this phenomenon and show that a criterion known as CAR (`coarsening at random') in the statistical literature characterizes when "naive" conditioning works. We provide two characterizations of CAR. First we show that in many situations, CAR essentially *cannot* hold, so that naive conditioning must give the wrong answer. This is problematic since people apply naive conditioning all the time. Second, we precisely characterize CAR by providing a randomized algorithm that can simulate all and only CAR distributions. This solves a tricky open problem first posed by Gill, van der Laan and Robins (1997).
The talk is based on joint work with R. Gill (Leiden) and J. Halpern (Cornell).
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