The author classifies the posterior distribution of a random density given a sample in the nonparametric context. Let K be a nonnegative kernel on the product of Borel subsets \({\mathcal X}\times {\mathcal R}\). Let \(\Theta\) be the set of all distributions on \({\mathcal R}\) and let \({\mathcal P}_{\alpha}\) be a Dirichlet probability on \(\Theta\) with a prescribed finite index measure \(\alpha\) on \({\mathcal R}\) (with suitable \(\sigma\) fields). For \(G\in \Theta\) the density \(f(x| G)=\int_{{\mathcal R}}K(x,u)G(du)\) is given. Let \(E^ X\) be the conditional expectation given the sample \(X_ 1,...,X_ n.\)
In theorem 1 the general form \(E^ Xg(G)\) for nonnegative or integrable functions g is given. In theorem 2 the conditional expectation \(E^ Xf(x| G)\) is computed. In the final part the author discusses the choice of K and \(\alpha\) and gives some examples.