Consistent semiparametric Bayesian inference about a location parameter

Abstract

We consider the problem of Bayesian inference about the centre of symmetry of a symmetric density on the real line based on independent identically distributed observations. A result of Diaconis and Freedman shows that the posterior distribution of the location parameter may be inconsistent if (symmetrized) Dirichlet process prior is used for the unknown distribution function. We choose a symmetrized Polya tree prior for the unknown density and independently choose θ according to a continuous and positive prior density on the real line. Suppose that the parameters of Polya tree depend only on the level m of the tree and the common values r_m's are such that ∑_m=1^∞ r_m^-½ < ∞. Then it is shown that for a large class of true symmetric densities, including the trimodal distribution of Diaconis and Freedman, the marginal posterior of θ is consistent.