A Fresh Look at Model Selection in Inverse Scaterring

Abstract

In this paper, we return to a problem which we treated at 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.

It is shown that one particular set of expansion functions is especially suitable. The Fourier-Bessel expansion can be derived from a variational principle involving the finite extent of nuclear charge and the sharp decline of its Fourier transform as a function of momentum transfer. We show expansions at different model orders and choose the one with the largest evidence.

The prior probability for the expansion coefficients is assigned by an application of Jaynes’ principle of maximum entropy. The parameter introduced to satisfy the assumed constraint in the data can be marginalized or estimated from the posterior distribution. Alternatively, we show that it can usefully be assigned directly from a single macroscopic variable derived from the data. This approach removes the need for an optimization and is conceptually simple. In addition, it gives results which are indistinguishable from those that estimate the constant a posteriori. The prior is shown to play a useful role in preventing the overfitting of data in an over-parametrized model.