Demystifying Dilation

Dilation occurs when upper and lower probability estimates of some event E are properly included in the upper and lower probability estimates of the probability of E conditional on another event F , resulting in a change from a more precise estimate of E to a less precise estimate of E upon learning F . Strict dilation occurs when E is dilated by every event in a partition, which means that sometimes E becomes less precise no matter how an experiment turns out. Many think that strict dilation is a pathological feature of imprecise probability models, while others have thought the problem is with Bayesian updating. However, a point often overlooked in critical discussions of dilation is that knowing that E is stochastically independent of F (for all F in a partition) is sufficient to avoid strict dilation. Since the most sensational alleged dilation examples are those which play up independence between dilator and dilatee, the sensationalism traces to mishandling imprecise probabilities rather than revealing a genuine puzzle about imprecise probabilities.