Functional Conjugacy in Parametric Bayesian Models

We address a basic question in Bayesian analysis: Can updates of the posterior under observations be represented as a closed-form mapping from the data to the posterior parameters? The question is closely related to the concept of a conjugate prior, but we do not assume that prior and posterior belong to the same model class. We refer to models for which a closed-form mapping exists as functionally conjugate, and ask which observation models admit such functionally conjugate priors. For finite-dimensional, dominated models, the answer is almost disappointingly restrictive: Under mild regularity assumptions, such a mapping can only exist if the likelihood is an exponential family model. This is a consequence of a more general result: In dominated models with strictly positive prior densities, existence of a mapping to the posterior parameters of the Bayesian model implies the existence of a sufficient statistic for the sampling model.