Updating Probabilities
This addresses foundational issues in probability theory and statistics, with implications for fields relying on probabilistic reasoning, though it is incremental as it generalizes and interconnects existing results.
The paper tackles the problem of updating probability distributions in 'naive spaces' where standard conditioning can lead to counterintuitive results, showing that the CAR condition for valid naive conditioning holds infrequently and that Jeffrey conditioning has a generalized condition while MRE does not.
As examples such as the Monty Hall puzzle show, applying conditioning to update a probability distribution on a ``naive space', which does not take into account the protocol used, can often lead to counterintuitive results. Here we examine why. A criterion known as CAR (coarsening at random) in the statistical literature characterizes when ``naive' conditioning in a naive space works. We show that the CAR condition holds rather infrequently. We then consider more generalized notions of update such as Jeffrey conditioning and minimizing relative entropy (MRE). We give a generalization of the CAR condition that characterizes when Jeffrey conditioning leads to appropriate answers, but show that there are no such conditions for MRE. This generalizes and interconnects previous results obtained in the literature on CAR and MRE.