Let $\mathscr{K} = \{K_\lambda: \lambda \in \Lambda\}$ be a family of sampling distributions for the data $x$ on a sample space $\mathscr{X}$ which is indexed by a parameter $\lambda \in \Lambda,$ and let $\mathscr{F}$ be a family of priors on $\Lambda$. Then $\mathscr{F}$ is said to be conjugate for $\mathscr{K}$ if it is closed under sampling, that is, if the posterior distributions of $\lambda$ given the data $x$ belong to $\mathscr{F}$ for almost all $x$. In this paper, we set up a framework for the study of what we term the dual problem: for a given family of priors $\mathscr{F}$ (a subfamily of a general exponential family), find the class of sampling models $\mathscr{K}$ for which $\mathscr{F}$ is conjugate. In particular, we show that $\mathscr{K}$ must be a general exponential family dominated by some measure $Q$ on $(\mathscr{X}, B),$ where $B$ is the Borel field on $\mathscr{X}$. It is the class of such measures $Q$ that we investigate in this paper. We study its geometric features and general structure and apply the results to some familiar examples.