On Nonparametric Bayesian Inference for the Distribution of a Random Sample

The nonparametric Bayesian approach for inference regarding the unknown distribution of a random sample customarily assumes that this distribution is random and arises through Dirichlet process mixing. Previous work within this setting has focused on the mean of the posterior distribution of this random distribution which is the predictive distribution of a future observation given the sample. Our interest here is in learning about other features of this posterior distribution as well as about posteriors associated with functionals of the distribution of the data. We indicate how to do this in the case of linear functionals. An illustration, with a sample from a Gamma distribution, utilizes Dirichlet process mixtures of normals to recover this distribution and its features.