Bayesian Nonparametric Inference of Decreasing Densities

In this paper we discuss consistency of the posterior distribution in cases where the Kullback-Leibler condition is not verified. This condition is stated as : for all $\epsilon > 0$ the prior probability of sets in the form $\{f ; KL(f0 , f ) \leq \epsilon\}$ where KL(f0 , f ) denotes the Kullback-Leibler divergence between the true density f0 of the observations and the density f , is positive. This condi- tion is in almost cases required to lead to weak consistency of the posterior distribution, and thus to lead also to strong consistency. However it is not a necessary condition. We therefore present a new condition to replace the Kullback-Leibler condition, which is usefull in cases such as the estimation of decreasing densities. We then study some specific families of priors adapted to the estimation of decreasing densities and provide posterior concentration rate for these priors, which is the same rate a the convergence rate of the maximum likelihood estimator. Some simulation results are provided.