A bayesian approach to the probability density estimation

Summary A Bayesian procedure for the probability density estimation is proposed. The procedure is based on the multinomial logit transformations of the parameters of a finely segmented histogram model. The smoothness of the estimated density is guaranteed by the introduction of a prior distribution of the parameters. The estimates of the parameters are defined as the mode of the posterior distribution. The prior distribution has several adjustable parameters (hyper-parameters), whose values are chosen so that ABIC (Akaike's Bayesian Information Criterion) is minimized. The basic procedure is developed under the assumption that the density is defined on a bounded interval. The handling of the general case where the support of the density function is not necessarily bounded is also discussed. The practical usefulness of the procedure is demonstrated by numerical examples. The Institute of Statistical Mathematics