Interpolation models with multiple hyperparameters

A traditional interpolation model is characterized by the choice of reg-ularizer applied to the interpolant, and the choice of noise model. Typically , the regularizer has a single regularization constant , and the noise model has a single parameter. The ratio == alone is responsible for determining globally all these attributes of the interpolant: its`complexity', ``exibility', `smoothness', `characteristic scale length', and`characteristic amplitude'. We suggest that interpolation models should be able to capture more than just one avour of simplicity and complexity. We describe Bayesian models in which the interpolant has a smoothness that varies spatially. We emphasize the importance, in practical implementation, of the concept of`conditional convexity' when designing models with many hyperparameters. We apply the new models to the interpolation of neuronal spike data and demonstrate a substantial improvement in generalization error.