Default Priors, Model Selection and Adaptation in Density Estimation

Default priors for density estimation may be constructed through certain approximating sieves with certain natural default priors on these sieves. Finite sieves with metric approximation properties and convolution-sieves provide natural examples and lead to consistent posteriors. Best rate of convergence may be achieved with more reened constructions of sieves using bracketing or spline functions, depending on the smoothness class of the densities. If the smoothness class is unknown, we consider hierarchical forms of the above priors arising from the uncertainty in the smoothness index and view the problem as model selection plus estimation. The posterior based on the hierarchical prior chooses less smooth classes with negligibly small probability. As a result, the resulting Bayes estimate is adaptive in the sense that the best rate of converegence is obtained for every smoothness class with the same prior.