We formulate the inverse problem of solving Fredholm integral equations of the first kind as a nonparametric Bayesian inference problem, using Lévy random fields (and their mixtures) as prior distributions. Posterior distributions for all features of interest are computed using novel Markov chain Monte Carlo numerical methods in infinite-dimensional spaces, based on generalizations and extensions of the authors' Inverse Lévy Measure (ILM) algorithm. The method is also well suited for deconvolution problems, for inverting Laplace and Fourier transforms, and for other linear and nonlinear problems in which the unknown feature is high- (or even infinite-) dimensional and where the corresponding forward problem may be solved rapidly.

The methods are illustrated in an application to an important problem in rheology: that of inferring the molecular weight distribution of polymers from conventional rheological measurements, in which we achieve not just a point estimate but a posterior probability density plot representing all uncertainty about the weight.

Keywords: Bayesian analysis; inverse problems; Lévy processes; nonparametrics; rheology.

The preprint (242 kb) and published manuscript (244 kb) are both available in PDF format. Cite as:

@Article{wolp:icks:2004,
      Author = "Robert L. Wolpert and Katja Ickstadt",
       Title = "Reflecting Uncertainty in Inverse Problems:
                  A {B}ayesian Solution using {L}\'{e}vy Processes",
     Journal = "Inverse Problems",
      Volume = 20,
      Number = 6,
       Pages = "1759--1771",
         DOI = "10.1088/0266-5611/20/6/004",
        Year = 2004,
}