Consider an estimate $\theta^*$ of a parameter $\theta$ based on
repeated observations from a family of densities $f_\theta$ evaluated by the
Kullback–Leibler loss function $K(\theta, \theta^*) = \int
\log(f_\theta/f_{\theta^*})f_\theta$. The maximum likelihood prior density, if
it exists, is the density for which the corresponding Bayes estimate is
asymptotically negligibly different from the maximum likelihood estimate. The
Bayes estimate corresponding to the maximum likelihood prior is identical to
maximum likelihood for exponential families of densities. In predicting the
next observation, the maximum likelihood prior produces a predictive
distribution that is asymptotically at least as close, in expected truncated
Kullback–Leibler distance, to the true density as the density indexed by
the maximum likelihood estimate. It frequently happens in more than one
dimension that maximum likelihood corresponds to no prior density, and in that
case the maximum likelihood estimate is asymptotically inadmissible and may be
improved upon by using the estimate corresponding to a least favorable prior.
As in Brown, the asymptotic risk for an arbitrary estimate “near”
maximum likelihood is given by an expression involving derivatives of the
estimator and of the information matrix. Admissibility questions for these
“near ML” estimates are determined by the existence of solutions
to certain differential equations.