12

Bayesian Splines

# 12.1 Introduction

Smoothing splines are an example of the class of functions called ‘scatterplot smoothers' which can be used in non-parametric regression. Smoothing splines are functions which are able to model the effect of a covariate on a response variable without assuming the functional form of the relationship or even that the effect of the covariate remains the same across the domain of the covariate. For this reason, smoothing splines have found application in generalized additive models (GAMs), an extension to the generalized linear model (GLM) (Hastie 1993; Wood 2006).

In section 12.2 we will give a brief introduction to some Bayesian statistical theory and the Metropolis–Hastings sampler, describe various spline models and discuss the interpretation of posterior summary statistics. In Section 12.3 we will give a number of examples of the use of smoothing splines within generalized additive models and use different data sets which call for different approaches. We will summarize the key points in Section 12.4 and discuss various ways in which the approach here can be extended.

# 12.2 Models and Methods

## 12.2.1 Splines and Linear Models

Splines are piecewise-defined functions constructed from simpler functions (usually polynomials) between control points (or ‘knots'). Splines have found ...