# Demystifying Dilation

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## Abstract

Dilation occurs when an interval probability estimate of some event *E* is properly included in the interval probability estimate of *E* conditional on every event *F* of some partition, which means that one’s initial estimate of *E* becomes less precise no matter how an experiment turns out. Critics maintain that dilation is a pathological feature of imprecise probability models, while others have thought the problem is with Bayesian updating. However, two points are often overlooked: (1) knowing that *E* is stochastically independent of *F* (for all *F* in a partition of the underlying state space) is sufficient to avoid dilation, but (2) stochastic independence is not the only independence concept at play within imprecise probability models. In this paper we give a simple characterization of dilation formulated in terms of deviation from stochastic independence, propose a measure of dilation, and distinguish between proper and improper dilation. Through this we revisit the most sensational examples of dilation, which play up independence between *dilator* and *dilatee*, and find the sensationalism undermined by either fallacious reasoning with imprecise probabilities or improperly constructed imprecise probability models.

## Notes

### Acknowledgments

Thanks to Horacio Arló-Costa, Jim Joyce, Isaac Levi, Teddy Seidenfeld, and Jon Williamson for their comments on early drafts, and to Clark Glymour and Choh Man Teng for a long discussion on dilation that began one sunny afternoon in a Lisbon café. This research was supported in part by award (LogICCC/0001/2007) from the European Science Foundation.

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