  We consider situations in Bayesian analysis where we have a family of priors
$\nu_h$ on the parameter $\theta$, where $h$ varies continuously over a space
$\mathcal{H}$, and we deal with two related problems. The first involves
sensitivity analysis and is stated as follows. Suppose we fix a function $f$ of
$\theta$. How do we efficiently estimate the posterior expectation of
$f(\theta)$ simultaneously for all $h$ in $\mathcal{H}$? The second problem is
how do we identify subsets of $\mathcal{H}$ which give rise to reasonable
choices of $\nu_h$? We assume that we are able to generate Markov chain samples
from the posterior for a finite number of the priors, and we develop a
methodology, based on a combination of importance sampling and the use of
control variates, for dealing with these two problems. The methodology applies
very generally, and we show how it applies in particular to a commonly used
model for variable selection in Bayesian linear regression, and give an
illustration on the US crime data of Vandaele.
