Bayesian curve-fitting with free-knot splines

We describe a Bayesian method, for fitting curves to data drawn from an exponential family, that uses splines for which the number and locations of knots are free parameters. The method uses reversible-jump Markov chain Monte Carlo to change the knot configurations and a locality heuristic to speed up mixing. For nonnormal models, we approximate the integrated likelihood ratios needed to compute acceptance probabilities by using the Bayesian information criterion, BIC, under priors that make this approximation accurate. Our technique is based on a marginalised chain on the knot number and locations, but we provide methods for inference about the regression coefficients, and functions of them, in both normal and nonnormal models. Simulation results suggest that the method performs well, and we illustrate the method in two neuroscience applications.