Abstract
We consider the problem of Bayesian density estimation on the positive semiline for possibly unbounded densities. We propose a hierarchical Bayesian estimator based on the gamma mixture prior which can be viewed as a location mixture. We study convergence rates of Bayesian density estimators based on such mixtures. We construct approximations of the local H\"older densities, and of their extension to unbounded densities, to be continuous mixtures of gamma distributions, leading to approximations of such densities by finite mixtures. These results are then used to derive posterior concentration rates, with priors based on these mixture models. The rates are minimax (up to a log n term) and since the priors are independent of the smoothness the rates are adaptive to the smoothness.
Highlights
Apart from the latter paper, the posterior concentration rates have been obtained by the above authors are equal to the minimax estimation rate over some collections of functional classes, showing that nonparametric mixture models are flexible prior models, but they lead to optimal procedures, in the frequentist sense
In this paper we propose to estimate a possibly unbounded density supported on the positive semiline via a Bayesian approach using a Dirichlet Process mixture of Gamma densities as a prior distribution
The main purpose of the paper is to derive the conditions on the Gamma mixture prior and on the hyperpriors so that the posterior distribution asymptotically concentrates at the optimal rate around the true density over smooth classes of densities
Summary
Nonparametric density estimation using Bayesian models with a mixture prior distribution has been used extensively in practice due to their flexibility and available computational techniques using MCMC. In some cases their theoretical properties have been studied, and in particular the asymptotic behaviour of the associated posterior distribution. The main purpose of the paper is to derive the conditions on the Gamma mixture prior and on the hyperpriors so that the posterior distribution asymptotically concentrates at the optimal rate (up to a log factor) around the true density over smooth classes of densities.
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