The maximum likelihood prior

Consider an estimate theta* of a parameter theta based on repeated observat ions from a family of densities f(theta) evaluated by the Kullback-Leibler loss function K(theta, theta*) = integral log(f(theta)/f(theta*))f(theta). The maximum likelihood prior density, if it exists, is the density for whic h the corresponding Bayes estimate is asymptotically negligibly different f rom the maximum likelihood estimate. The Bayes estimate corresponding to th e maximum likelihood prior is identical to maximum likelihood for exponenti al families of densities. In predicting the next observation, the maximum l ikelihood prior produces a predictive distribution that is asymptotically a t least as close, in expected truncated Kullback-Leibler distance, to the t rue density as the density indexed by the maximum likelihood estimate. It f requently happens in more than one dimension that maximum likelihood corres ponds to no prior density, and in that case the maximum likelihood estimate is asymptotically inadmissible and may be improved upon by using the estim ate corresponding to a least favorable prior. As in Brown, the asymptotic r isk for an arbitrary estimate "near ML" maximum likelihood is given by an e xpression involving derivatives of the estimator and of the information mat rix. Admissibility questions for these "near ML" estimates are determined b y the existence of solutions to certain differential equations.