We give an upper bound for the posterior probability of a measurable set $A$ when the prior lies in a class of probability measures $\mathscr{P}$. The bound is a rational function of two Choquet integrals. If $\mathscr{P}$ is weakly compact and is closed with respect to majorization, then the bound is sharp if and only if the upper prior probability is 2-alternating. The result is used to compute bounds for several sets of priors used in robust Bayesian inference. The result may be regarded as a characterization of 2-alternating Choquet capacities.
@article{1176347752,
author = {Wasserman, Larry A. and Kadane, Joseph B.},
title = {Bayes' Theorem for Choquet Capacities},
journal = {Ann. Statist.},
volume = {18},
number = {1},
year = {1990},
pages = { 1328-1339},
language = {en},
url = {http://dml.mathdoc.fr/item/1176347752}
}
Wasserman, Larry A.; Kadane, Joseph B. Bayes' Theorem for Choquet Capacities. Ann. Statist., Tome 18 (1990) no. 1, pp. 1328-1339. http://gdmltest.u-ga.fr/item/1176347752/