Safe Probability: Restricted Conditioning and Extended Marginalization

Updating probabilities by conditioning can lead to bad predictions, unless one explicitly takes into account the mechanisms that determine (1) what is observed and (2) what has to be predicted. Analogous to the observation-CAR (coarsening at random) condition, used in existing analyses of (1), we propose a new prediction task-CAR condition to analyze (2). We redefine conditioning so that it remains valid if the mechanisms (1) and (2) are unknown. This will often update a singleton distribution to an imprecise set of probabilities, leading to dilation, but we show how to mitigate this problem by marginalization. We illustrate our notions using the Monty Hall Puzzle.