# Maximum Entropy in Small-Angle Scattering

## Steen Hansen

Department of Mathematics and Physics

Royal Veterinary and Agricultural University

Thorvaldsensvej 40, 1871 FRB C, Denmark

### Abstract

Among its many possible applicatations, small-angle scattering is used
for obtaining structural information about molecules in solution. The
interest for performing solution experiments appears for example in
biophysics, where it may be crucial to preserve the exact and
functionally active structure of the biomolecule. The loss of
information owing to the random orientation of the molecules in the
solution makes it important to extract the maximum information from
the measured scattering profile.
For interpretation of the experimental results it is often relevant to
represent the scattering data in direct space, which requires a
Fourier transform of the experimental data. A direct Fourier transform
is of limited use, owing to noise, smearing and truncation. By
indirect Fourier transformation some of the problems can be overcome,
but the indirect transformation is an underdetermined problem
requiring regularisation to be used. MaxEnt is ideally suited for such
problems as it stabilizes the solution by allowing prior information
to be included in a logical and transparent way. Calculating the
number of good directions measured in a small-angle scattering
experiment shows the need for additional knowledge to be used in the
analysis of the data.

For small-angle scattering prior information about the structure of
the molecule usually will be available from other techniques as e.g.
electron microscopy.

Examples are given demonstrating the performance of MaxEnt compared to
some of the various other methods used for indirect Fourier
transformation in small-angle scattering (*ad hoc* smoothness
constraints (the method most frequently used) decomposition in exotic
functional systems and Tikhonov regularisation). It is shown how the
choice of prior influences the solution, and given the choice between
several models how the ``best'' can be chosen by maximization of the
entropy (in particular, any unknown parameter in the prior (e.g.
the overall size of the molecule) can be found by simultaneous
maximization of the entropy of the distribution to be estimated and
this parameter).

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