Maximum entropy method and Bayesian probability theory

The Maximum Entropy ($\textit{MaxEnt}$) method is a relatively new technique especially suitable for reconstructing positive and additive distributions from incomplete and noisy data. Such positive and additive distributions ($\textit{PAD}$) occur in many fields of science and engineering. For instance, the neutron scattering law $\textit{S(Q,E)}$ for a given sampie is a positive and additive distribution. Additional examples are: probability distributions, intensity of light as a function of position in an optical image, the intensity of radio signals from distant stars, power spectra, etc., to name only few. The success of MaxEnt method in the wide range of scientific and engineering fields is the consequence of its foundation in the $\textit{Bayesian}$ probability theory, which as will be shown in subsequent chapters, provides the logical and consistent way of reasoning in the presence of uncertainty. This fact is of great importance, especially in the solution of so called $\textit{inverse}$ problems. We show that such problems can be treated by means of probabilistic reasoning and inference founded in the $\textit{Bayes' rule}$. This approach to inverse problems is much more successful than the approach offered by mathematical inversion