Curve and Surface Estimation using Dynamic Step Functions

This chapter describes a nonparametric Bayesian approach to the estimation of curves and surfaces that act as parameters in statistical models. The approach is based on mixing variable dimensional piecewise constant approximations, whose ‘smoothness’ is regulated by a Markov random field prior. Random partitions of the domain are defined by Voronoi tessellations of random generating point patterns. Variable dimension Markov chain Monte Carlo methods are proposed for the numerical estimation, and a detailed algorithm is specified for one special case. General applicability of the approach is discussed in the context of density estimation, regression and interpolation problems, and an application to the intensity estimation for a spatial Poisson point process is presented.