Function estimation with locally adaptive dynamic models

We present a nonparametric Bayesian method for fitting unsmooth and
highly oscillating functions, which is based on a locally adaptive
hierarchical extension of standard dynamic or state space models.
The main idea is to introduce locally varying variances in the
state equations and to add a further smoothness prior for this
variance function. Estimation is fully Bayesian and carried out by
recent MCMC techniques. The whole approach can be understood as an
alternative to other nonparametric function estimators, such as
local or penalized regression with variable bandwidth or smoothing
parameter selection. Performance is illustrated with simulated
data, including unsmooth examples constructed for wavelet
shrinkage, and by an application to sales data. Although the
approach is developed for classical Gaussian nonparametric
regression, it can be extended to more complex regression problems.