  We give an exact characterization of admissibility in statistical decision
problems in terms of Bayes optimality in a so-called nonstandard extension of
the original decision problem, as introduced by Duanmu and Roy. Unlike the
consideration of improper priors or other generalized notions of Bayes
optimalitiy, the nonstandard extension is distinguished, in part, by having
priors that can assign "infinitesimal" mass in a sense that can be made
rigorous using results from nonstandard analysis. With these additional priors,
we find that, informally speaking, a decision procedure $\delta_0$ is
admissible in the original statistical decision problem if and only if, in the
nonstandard extension of the problem, the nonstandard extension of $\delta_0$
is Bayes optimal among the extensions of standard decision procedures with
respect to a nonstandard prior that assigns at least infinitesimal mass to
every standard parameter value. We use the above theorem to give further
characterizations of admissibility, one related to Blyth's method, one to a
condition due to Stein which characterizes admissibility under some regularity
assumptions; and finally, a characterization using finitely additive priors in
decision problems meeting certain regularity requirements. Our results imply
that Blyth's method is a sound and complete method for establishing
admissibility. Buoyed by this result, we revisit the univariate two-sample
common-mean problem, and show that the Graybill--Deal estimator is admissible
among a certain class of unbiased decision procedures.
