Bayesian approach to the choice of smoothing parameter in kernel density estimation

In data driven bandwidth selection procedures for density estimation such as least squares cross validation and biased cross validation, the choice of a single global bandwidth is too restrictive. It is however reasonable to assume that the bandwidth has a distribution of its own and that locally, depending on the data, the-bandwidth may differ. In this approach, the bandwidth is assigned a prior distribution in the neighborhood around the point at which the density is being estimated. Assuming that the kernel function is a proper probability distribution, a Bayesian approach is employed to come up with a posterior type distribution of the bandwidth given the data. Finally, the mean of the posterior distribution is used to select the local bandwidth.