## On a class of Bayesian nonparametric estimates: I. Density estimates.(English)Zbl 0557.62036

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.
Reviewer: P.Kischka

### MSC:

 62G05 Nonparametric estimation 62C10 Bayesian problems; characterization of Bayes procedures 62A01 Foundations and philosophical topics in statistics
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