Bayesian Spline Regression when the Number of Knots is Unknown

Summary The problem of selecting the model and coefficients for a spline regression curve with normal errors is considered. The locations of all possible knots are assumed to be known; which subset of these is the set of actual knots is unknown. Each subset forms a model. Priors of the form of a marginal distribution on the index of the model and a conditional distribution for each model on the coefficients and variance are allowed. The posterior distribution is explicitly derived for natural conjugate and vague priors. The optimal predictor is derived for a loss function which is a generalization of that appearing in Lindley (1968). All that is required of the opinion is the marginal distribution for the model and the mean vector for the coefficients.