Consistency of the posterior distribution without the Kullback-Leibler condition

Consistency of the posterior distribution without the Kullback-Leibler condition

In this work, we study consistency in nonparametric Bayesian estimation of a nonincreasing density on R. Since such a density can be written a8 a. mixture of Uniform den it is natural to consider Bayesian procedures that are based on nonparametric mixture models. we consider in particular as priors the Dirichlet mixture process and finite mixtures with unknown number of components. We show that for finite mixture priors. the Kullback-Leibler property is not satisfied, and we propose an alternative condition that still ensures consistency of the posterior distributions and which is satisfied in those models. As an application of the main theorem we define the nonincreasing densities on 0.1 and we proof the posterior distribution is consistent. This result can be extended to.