A bayesian analysis for less smooth departures from polynomial regression models

Abstract

Observations y_i are made at points t_i according to the model _i=θ(t_i)+e_i where the e_i are independent normals with constant variance. It is conjec tured that the function θ lies in a set G of functions spanned by the basis functions φ₁,φ₂,…,φ_p. A prior distribution is developed whereby the proba bility assigned to a function θ is a decreasing function of a particular measure of the distance of θ from the set G. Bayes' theorem is used to construct an estimateθ. A marginal likelihood is derived which is used to estimate the parameter of the prior and also for testing the null hypothesis H_o θ ∊ G. The new methodology is tested in a Monte Carlo study and applied to a set of data representing the average weight to height ratio of a group of boys recorded at one month intervals.

Keywords:

• Bayesian analysts: smoothing splines
• generalized maximum likelihood
• basis functions