Conditionally reducible natural exponential families and enriched conjugate priors

Consider a standard conjugate family of prior distributions for a vectorparameter indexing an exponential family. Two distinct model parameterizations may well lead to standard conjugate families which are not consistent, i.e. one family cannot be derived from the other by the usual change-ofvariable technique. This raises the problem of finding suitable parameterizations that may lead to enriched conjugate families which are more flexible than the traditional ones. The previous remark motivates the definition of a new property for an exponential family, named conditional reducibility. Features of conditionally reducible natural exponential families are investigated thoroughly. In particular, we relate this new property to the notion of cut, and show that conditionally-reducible families admit a reparameterization in terms of a vector having likelihood-independent components. A general methodology to obtain enriched conjugate distributions for conditionally-reducible families is described in detail, generalizing previous works and more recent contributions in the area. The theory is illustrated with reference to natural exponential families having simple quadratic variance function.