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

Abstract

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.

Refinement of Macromolecular Structures by the Maximum-Likelihood Method

“…Then (9) will become the familiar Rice distribution (see Bricogne, 1988;Rice, 1954), This equation is the most popular one from the family of distributions given by (9). In crystallography a special case of this equation was first derived by Srinivasan & Ramachandran (1965).…”

“…The resulting marginal distribution of amplitudes of structure factors was given by Srinivasan & Ramachandran (1965). Later Bricogne (1988) noted that this distribution was used by other authors, the first being Rice (1954). Luzzati's results were extended by Read (1990) to include the assumption that there are errors in the atomic scattering factors which can be assigned to AB.…”

“…Recent interest in using these equations for refinement of macromolecular structures has been stimulated by their successful application in reducing model bias in map calculations (Lunin & Urzhumtsev, 1984;Read, 1986); by the theoretical results described by Bricogne (Bricogne, 1988) for their application with maximum entropy as a general tool in phase determination; by their successful application of his results in phase improvement (Carter, Crumley, Coleman, Hage & Bricogne, 1990); and by the increasing power of computers. The procedure described here is referred to as 'maximum-likelihood refinement' since all previous macromolecular refinement procedures are referred to as 'least-squares refinement' despite the fact that all these also utilize prior information.…”

The Influence of the Completeness of the Data Set on the Charge Density Obtained with the Maximum-Entropy Method. A Re-examination of the Electron-Density Distribution in Si

“…The maximum-entropy method (MEM) is a statistical deduction (Bricogne, 1988) that can yield a highresolution density distribution from a limited number of diffraction data without using a structural model (Collins, 1982;Sakata & Sato, 1990). It has been suggested that the MEM would be a suitable method for examining electron densities in the interatomic region, e.g.…”

Statistical pattern recognition for macromolecular crystallographers

“…Indeed, statistical pattern recognition may be seen as just another instance of applied Bayesian statistics, the use of which has been advocated for years by Jaynes (see references in Jaynes, 2003) and in crystallography by Bricogne and others (French & Wilson, 1978;French & Oatley, 1978;Bricogne, 1988Bricogne, , 1997a.…”