A Bayesian statistical theory of the phase problem. I. A multichannel maximum-entropy formalism for constructing generalized joint probability distributions of structure factors

In this first of three papers on a full Bayesian theory of crystal structure determination, it is shown that all currently used sources of phase information can be represented and combined through a universal expression for the joint probability distribution of structure factors. Particular attention is given to situations arising in macromolecular crystallography, where the proper treatment of non-uniform distributions of atoms is absolutely essential. A procedure is presented, in stages of gradually increasing complexity, for constructing the joint probability distribution of an arbitrary collection of structure factors. These structure factors may be gathered from one or several crystal forms of an unknown molecule, each comprising one or several isomorphous structures related by substitution operations, possibly containing solvent regions and known fragments, and/or obeying a set of non-crystallographic symmetries. This universal joint probability distribution can be effectively approximated by the saddlepoint method, using maximum-entropy distributions of atoms [Bricogne (1984). Acta Cryst. A40, 410-445] and a generalization of structure-factor algebra. Atomic scattering factors may assume arbitrary complex values, so that this formalism applies to neutron as well as to X-ray diffraction methods. This unified procedure will later be extended by the construction of conditional distributions allowing phase extension, and of likelihood functions capable of detecting and characterizing all potential sources of phase information considered so far, thus completing the formulation of a full Bayesian inference scheme for crystal structure determination.