University of Washington - Applied Physics Lab
Deconvolution of an unknown function of one variable from a finite set of measurements is an ill-posed problem. Placing a Bayesian prior on a function space is one way to extend the scientific model and obtain a well-posed problem. This problem can be well-posed even if the relationship between the unknown function and the measurements, as well as the function space prior, has unknown parameters. We present a method for estimating the unknown parameters by maximizing an approximation of the marginal likelihood where the unknown function has been integrated out. This is an extension of the marginal likelihood estimator for the regularization parameter because we allow for a nonlinear relationship between the unknown function and the measurements. The unknown function is estimated by maximizing its likelihood given the parameters and the data. We present a computational method that uses eigenfunctions to represent the smoothing spline. The importance of allowing for a nonlinear transformation is demonstrated by a stochastic sum of exponentials example.