Selfinformative Limits of Bayes Estimates and Generalized Maximum Likelihood

A definition of selfinformative Bayes carriers or limits is given as a description of an approach to noninformative Bayes estimation in non- and semiparametric models. It takes the posterior w.r.t. a prior as a new prior and repeats this procedure again and again. A main objective of the paper is to clarify the relation between selfinformative carriers or limits and maximum likelihood estimates (MLE's). For a model with dominated probability distributions we state sufficient conditions under which the set of MLE's is a selfinformative carrier or in the case of a unique MLE its selfinformative limit property. Mixture models are covered. The result on carriers is extended to more general models without dominating measure. Selfinformative limits in the case of estimation of hazard functions based in censored observations and in the case of normal linear models with possibly nonidentifiable parameters are shown to be identical to the generalized MLE's in the sense of Gill (1989) and Kiefer and Wolfowitz (1956). Selfinformative limits are given for semiparametric linear models. For a location model they are identical to generalized MLE's, while this is not true in general.