A1483
Title: Recurrence and transience of Markov chains and evaluation of improper priors
Authors: Kshitij Khare - University of Florida (United States) [presenting]
Abstract: Sufficient conditions are developed for recurrence and transience of irreducible Markov chains on the real line. The results are developed using a combination of drift (Lyapunov) conditions and increment analysis, which is based on the moments of the (one-step) jumps of the chain. The new results are applied to a simple, but fundamental problem in statistical decision theory. Specifically, suppose $U\sim N(t, 1)$ and let $a(t)$ be an improper prior density that yields a proper posterior density, $p(t|U=u)$. Consider the Markov chain on the real line whose one-step transition from x is obtained by first drawing $t\sim p(.|x)$, and then adding a standard normal noise to $t$. Previous results in the literature imply that, if this Markov chain is (null) recurrent, then the prior $a(t)$ is strongly admissible, which basically means that the Bayes estimators generated by $p(t|u)$ are admissible. The new sufficient conditions for recurrence are used to show that various improper priors for $t$, including improper versions of popular shrinkage priors, are strongly admissible.
