Abstract
A density bounded class P of probability distributions on a space χ is the set of all probability distributions corresponding to probability densities bounded below by a given subprob-ability density and bounded above by a given superprobability density. Density bounded classes arise in robust Bayesian analysis (Lavine 1991) and also in Monte Carlo integration (Fishman Granovsky and Rubin 1989). Finding upper and lower bounds on the variance over all p∊ P allows one to bound the Monte Carlo variance. Fishman Granovsky and Rubin (1989) find bounds on the variance over all p ∊ P and also find the densities in P achieving those bounds in the case where χ is discrete; that is, where P is actually a set of probability mass functions. This article generalizes their result by showing how to bound the variance and find the densities achieving the bounds when χ is continuous.
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