Conjugate Priors for Exponential Families

Let X be a random vector distributed according to an exponential family with natural parameter θ ∈ Θ. We characterize conjugate prior measures on Θ through the property of linear posterior expectation of the mean parameter of X : E(X|θ)|X = x\ = ax + b. We also delineate which hyperparameters permit such conjugate priors to be proper.