A fresh look at model selection in inverse scattering

V. A. Macaulay and B. Buck
Theoretical Physics
University of Oxford
1 Keble Road
Oxford OX1 3NP
United Kingdom
buck@thphys.ox.ac.uk and vincent@thphys.ox.ac.uk


In this talk, we return to the problem which we treated in MaxEnt89 and show how the Bayesian evidence formalism provides new insights. The problem is to infer the charge distribution of a nucleus from noisy and incomplete measurements of its Fourier transform. This topic is rather specialized, but we emphasize that features of the analysis are more generally applicable.

It is shown that a unique set of expansion functions (Fourier-Bessel functions) can be generated from a variational principle involving the finite extent of nuclear charge and the sharp decline of its Fourier transform as a function of wavevector. Others have made heuristic arguments to determine how many of these functions can be justified in the model. We use the Bayesian `evidence' formalism to determine this number. The prior probability for the expansion coefficients is generated with an application of Jaynes' principle of maximum entropy.

Much of the complexity of parameter fitting and model selection arises in trying to determine the hyperparameters that occur in the prior probabilities. These are usually either marginalized or estimated from the posterior distribution. We show how the regularizing parameter in this problem can usefully be assigned from a single macroscopic variable derived from the data. The method requires that we allow ourselves a coarse look at the data in order to determine that variable, which we take to be the power of the signal. This apporach removes the need for lengthy calculation and is conceptually simple. In addition, it gives results which are indistinguishable from those derived by estimating the regularization constant a posteriori; we try to show why this should be so. We will also compare the above discrete model with a free-form MaxEnt reconstruction of the same charge density. These ideas can be generalized to other models, such as discrete and free-form spectral bases. We conclude with a discussion of the implications of our work for other such analyses.

MaxEnt 94 Abstracts / mas@mrao.cam.ac.uk