The Annals of Statistics

Dilation for Sets of Probabilities

Teddy Seidenfeld and Larry Wasserman

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Suppose that a probability measure $P$ is known to lie in a set of probability measures $M$. Upper and lower bounds on the probability of any event may then be computed. Sometimes, the bounds on the probability of an event $A$ conditional on an event $B$ may strictly contain the bounds on the unconditional probability of $A$. Surprisingly, this might happen for every $B$ in a partition $\mathscr{B}$. If so, we say that dilation has occurred. In addition to being an interesting statistical curiosity, this counterintuitive phenomenon has important implications in robust Bayesian inference and in the theory of upper and lower probabilities. We investigate conditions under which dilation occurs and we study some of its implications. We characterize dilation immune neighborhoods of the uniform measure.

Article information

Ann. Statist., Volume 21, Number 3 (1993), 1139-1154.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62F15: Bayesian inference
Secondary: 62F35: Robustness and adaptive procedures

Conditional probability density ratio neighborhoods $\varepsilon$-contaminated neighborhoods robust Bayesian inference upper and lower probabilities


Seidenfeld, Teddy; Wasserman, Larry. Dilation for Sets of Probabilities. Ann. Statist. 21 (1993), no. 3, 1139--1154. doi:10.1214/aos/1176349254.

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