Abstract: The use of a finite dimensional Dirichlet prior in the finite normal mixture model has the effect of acting like a Bayesian method of sieves. Posterior consistency is directly related to the dimension of the sieve and the choice of the Dirichlet parameters in the prior. We find that naive use of the popular uniform Dirichlet prior leads to an inconsistent posterior. However, a simple adjustment to the parameters in the prior induces a random probability measure that approximates the Dirichlet process and yields a posterior that is strongly consistent for the density and weakly consistent for the unknown mixing distribution. The dimension of the resulting sieve can be selected easily in practice and a simple and efficient Gibbs sampler can be used to sample the posterior of the mixing distribution. Key words and phrases: Bose-Einstein distribution, Dirichlet process, identification, method of sieves, random probability measure, relative entropy, weak convergence.

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We consider multivariate density estimation with identically distributed observations. We study a density estimator which is a convex combination of functions in a dictionary and the convex combination is chosen by minimizing the L2 empirical risk in a stagewise manner. We derive the convergence rates of the estimator when the estimated density belongs to the L2 closure of the convex hull of a class of functions which satisfies entropy conditions. The L2 closure of a convex hull is a large non-parametric class but under suitable entropy conditions the convergence rates of the estimator do not depend on the dimension, and density estimation is feasible also in high dimensional cases. The variance of the estimator does not increase when the number of components of the estimator increases. Instead, we control the bias-variance trade-off by the choice of the dictionary from which the components are chosen.

Discrete dynamic models: