## The Annals of Statistics

### Dilation for Sets of Probabilities

#### Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176349254

Digital Object Identifier
doi:10.1214/aos/1176349254

Mathematical Reviews number (MathSciNet)
MR1241261

Zentralblatt MATH identifier
0796.62005

JSTOR