Rates of convergence for the posterior distributions of mixtures of Betas and adaptive nonparametric estimation of the density

In this paper, we investigate the asymptotic properties of nonparametric Bayesian mixtures of Betas for estimating a smooth density on $[0,1]$. We consider a parametrization of Beta distributions in terms of mean and scale parameters and construct a mixture of these Betas in the mean parameter, while putting a prior on this scaling parameter. We prove that such Bayesian nonparametric models have good frequentist asymptotic properties. We determine the posterior rate of concentration around the true density and prove that it is the minimax rate of concentration when the true density belongs to a H\"{o}lder class with regularity $\beta$, for all positive $\beta$, leading to a minimax adaptive estimating procedure of the density. We also believe that the approximating results obtained on these mixtures of Beta densities can be of interest in a frequentist framework.Comment: Published in at http://dx.doi.org/10.1214/09-AOS703 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Some remarks on the continuity equation

We describe some relations between the properties of the Cauchy problem for an ODE and the properties of the Cauchy problem for the associated continuity equation in the class of measures

License GPL (> = 2) Repository CRAN Date/Publication 2009-04-28 10:10:13

Description mcsm contains a collection of functions that allows the reenactment of the R programs used in the book EnteR Monte Carlo Methods without further programming. Programs being available as well, they can be modified by the user to conduct one’s own simulations