Bayesian spline smoothing with ambiguous penalties

Abstract

A popular method for flexible function estimation in nonparametric models is the smoothing spline. When applying the smoothing spline method, the nonparametric function is estimated via penalized least squares, where the penalty imposes a soft constraint on the function to be estimated. The specification of the penalty functional is usually based on a set of assumptions about the function. Choosing a reasonable penalty function is the key to the success of the smoothing spline method. In practice, there may exist multiple sets of widely accepted assumptions, leading to different penalties, which then yield different estimates. We refer to this problem as the problem of ambiguous penalties. Neglecting the underlying ambiguity and proceeding to the model with one of the candidate penalties may produce misleading results. In this article, we adopt a Bayesian perspective and propose a fully Bayesian approach that takes into consideration all the penalties as well as the ambiguity in choosing them. We also propose a sampling algorithm for drawing samples from the posterior distribution. Data analysis based on simulated and real‐world examples is used to demonstrate the efficiency of our proposed method.