We consider situations in Bayesian analysis where we have a family of priors $ν_h$ on the parameter $θ$, 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 $θ$. How do we efficiently estimate the posterior expectation of $f(θ)$ 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 $ν_h$? 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.