Conditionally reducible natural exponential families and enriched conjugate priors

Consider a standard conjugate family of prior distributions for a vector-pa rameter indexing an exponential family. Two distinct model parameterization s 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-of-varia ble technique. This raises the problem of finding suitable parameterization s that may lead to enriched conjugate families which are more flexible than the traditional ones. The previous remark motivates the definition of a ne w property for an exponential family, named conditional reducibility, Featu res of conditionally-reducible natural exponential families are investigate d thoroughly. In particular, we relate this new property to the notion of c ut, and show that conditionally-reducible families admit a reparameterizati on in terms of a vector having Likelihood-independent components. A general methodology to obtain enriched conjugate distributions for conditionally-r educible families is described in detail, generalizing previous works and m ore recent contributions in the area, The theory is illustrated with refere nce to natural exponential families having simple quadratic variance functi on.