Estimating parameters and stochastic functions of one variable using nonlinear measurements models

Estimating an unknown function of one variable from a finite set of measurements is an ill-posed inverse problem. Placing a Bayesian prior on a function space is one way to make this problem well-posed. This problem can turn out well-posed even if the relationship between the unknown function and the measurements, as well as the function space prior, contains 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 marginal likelihood estimators for the regularization parameter because we allow for a nonlinear relationship between the unknown function and the measurements. The estimate of the function is then obtained by maximizing its a posteriori probability density function given the parameters and the data. We present a computational method that uses eigenfunctions to represent the function space. The continuity properties of the function estimate are characterized. Proofs of the convergence of the method are included. The importance of allowing for a nonlinear transformation is demonstrated by a stochastic sum of exponentials example.