The maximum-entropy method without the positivity constraint - applications to the determination of the distance-distribution function in small-angle scattering

The maximum-entropy principle (MaxEnt) is well suited to the solution of underdetermined problems by inclusion of prior knowledge in a logically consistent way. In most applications of MaxEnt, a set of numbers - pixel densities in an image or counts in a spectrum - are determined. In these cases, the set of numbers can be interpreted as a probability distribution and is, as such, all positive. It is the purpose here to show that MaxEnt is able to provide estimates of a set of quantities that cannot necessarily be interpreted as probabilities and that may become negative, as is the case for the correlation function in small-angle scattering. The method is illustrated both by analysis of simulated data and by measurements on sodium dodecylsulfate.