Resolution and Scale Independent Function Matching Using a String Energy Penalized Spline Prior

TL;DR

This paper extends Bayesian penalized spline methods to vector-valued functions, introducing a scale-independent approach inspired by string energy physics, with applications to multidimensional numerical integrators.

Contribution

It proposes a novel scale independence criterion for penalty parameter selection and introduces a string zero-point energy to address polynomial fit issues.

Findings

Method achieves resolution independence in function estimates.

Introduces a new Bayesian penalty parameter selection approach.

Demonstrates effectiveness on stochastic numerical integrators.

Abstract

The extension of the classical Bayesian penalized spline method to inference on vector-valued functions is considered, with an emphasis on characterizing the suitability of the method for general application.We show that the standard quadratic penalty is exactly analogous to the energy of a stretched string, with the penalty parameter corresponding to its tension. This physical analogy motivates a discussion of resolution independence, which we define as the convergence of a computational function estimate to arbitrary accuracy with increasing resolution.The multidimensional context makes direct application of standard procedures for choosing the penalty parameter difficult, and a new method is proposed and compared to the established generalized cross-validation (GCV) and Akaike information criterion (AIC) functions.Our Bayesian method for choosing this parameter is derived by introducing a scal e independence criterion to ensure that simultaneously scaling the function samples and their variances does not significantly change the posterior parameter distribution. Due to the possibility of an exact polynomial fit, numerical issues prevent the use of this prior, and a solution is presented based on adding a st ring zero-point energy. This makes more complicated approaches recently propose d in the literature unnecessary, and eliminates the requirement for sensitivity analysis when the function deviates from the above mentioned polynomial. An important class of problems which can be analyzed by this method are stochastic numerical integrators, which are considered as an example problem. This work represents the first extension of penalized spline methods to inference on multidimensional numerical integrators reported in the literature. Several numerical calculations illustrate the above points and address practical application issues.