  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.
