Discrete noninformative priors

There are various rules to construct "noninformative priors" in Bayesian analysis. We propose noninformative priors to be least favorable in a certain game: the nature space is a convex hull of densities; the action space is the collection of all density functions; and the loss function is the Kullback-Leibler distance. When the convex hull is generated by a parametric family of densities of a random sample $X\sb1, X\sb2,\cdot\cdot\cdot, X\sb{n}$ on a compact parameter space, the asymptotic least favorable prior, as $n \to\ \infty$, is the Berger-Bernardo prior. For one observation X sampled from a location family, we study the asymptotic behavior of least favorable priors on a sequence of nested intervals when they extend either to the half line or to the whole line. The asymptotic least favorable prior for either the uniform location family or the normal location family on the half line is shown to be a unique, discrete, improper distribution. For the whole line, the asymptotic least favorable prior for the uniform density is discrete but not unique. However, for the normal $N(\theta$, 1) with $\theta\ \in\ {\cal R}$, the Jeffreys prior is the unique asymptotic least favorable prior.