ABSTRACT
Smoothing methods have been playing an important role in the nonparametric approach to regression. This chapter introduces a few smoothing models that have been extensively used in statistical fields. It demonstrates how these models are linked to random walk priors under the Bayesian framework, and how to make Bayesian inference on those models using integrated nested Laplace approximations in simulated and real data examples. It fits a low-rank thin-plate spline, which is constructed by starting with the basis and penalty for a full thin-plate spline, and then truncating this basis in an optimal manner to obtain a low rank smoother. Since some priors mentioned have improper densities, the corresponding joint posterior distribution may be improper as well, which leads to invalid Bayesian inference. The German tenancy law puts restrictions on the increase of rents and forces landlords to keep the price in a range defined by apartments which are comparable in size, location and quality.
