Selection models arise when the data is selected to enter the sample only if it occurs in a certain region of the sample space. When this selection occurs according to some probability distribution, the resulting model is often instead called a weighted distribution model. In either case the ``original'' density becomes multiplied by a ``weight function'' $w(x)$. Often there is considerable uncertainty concerning this weight function; for instance, it may only be known that $w$ lies between two specified weight functions. We consider robust Bayesian analysis for this situation, finding the range of posterior quantities of interest, such as the posterior mean or posterior probability of a set, as $w$ ranges over the class of weight functions. The variational analysis utilizes concepts from variation diminishing transformations. PDF File