A Statistical Diptych: Admissible Inferences--Recurrence of Symmetric Markov Chains

Given a parametric model and an improper prior distribution, the formal posterior distribution induces decision rules in any decision problem. The results here provide conditions under which this formal Bayes method produces admissible decision rules for all quadratically regular decision problems. The conditions derived are shown to be equivalent to the recurrence of a natural symmetric Markov chain (on the parameter space) generated by the model and the improper prior. The results are also used to give conditions under which formal predictive distributions are admissible decision rules in certain prediction problems.