# 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`

### Abstract

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