Use of Statistical Information in X-ray Crystallography with Application to the Holographic Method

The holographic method solves the crystallographic inverse problem in real space. In addition to the measured structure-factor amplitudes, it uses other available information such as the positivity of the electron density, knowledge of part of the structure as well as MIR and/or MAD data. In the present paper, the range of useful information is extended to include knowledge that is statistical in nature. For example, it is known that the distribution of the structure-factor amplitudes of large molecules is described by Wilson statistics. Bayesian methods are used to optimize the signal-to-noise ratio of experimental measurements, to estimate missing reflections and to extrapolate measured data to higher resolution. In a similar vein, the cost function in the holographic algorithm is modified to account for the uncertainties of the measured structure factors. It is also shown how statistical knowledge about the unsolved part of the molecule may be utilized.