A multisolution method of phase determination by combined maximization of entropy and likelihood. I. Theory, algorithms and strategy

A new multisolution method for direct phase determination [Bricogne (1984). Acta Cryst. A40, 410-445] has been implemented and tested on small crystal structures. It consists of an organized search for those combinations of phases associated with a 'basis set' of reflexions which have maximum likelihood, i.e. which lead to the assignment of the highest conditional probability to the observed moduli belonging to reflexions outside the basis set. Phase choices are made sequentially, progressively enlarging the basis set, and the book-keeping involves a 'multisolution tree' which summarizes the parentage relations between them. The conditional probability distributions (c.p.d.'s) of structure factors used in evaluating the likelihood are derived from joint distributions obtained by the saddlepoint method. The latter involves distributions of atoms which have the maximum entropy compatible with all phase choices made, and hence are different for each node of the multisolution tree. These distributions q^ME are constructed numerically by exponentially modelling, coupled with a very robust plane search which often simplifies to a line search. C.p.d.'s of small numbers of structure factors not in the basis set are readily calculated from q^ME, with correct representation of their multimodality. A further 'diagonal' approximation of these c.p.d.'s allows the log-likelihood to be written as a sum of contributions from individual non-basis reflexions. The phasing process is initiated by specifying the origin-fixing and enantiomorph-defining phases, and forming the corresponding q^ME. It progresses by roughly locating the maxima of the c.p.d.'s of additional structure factors by a magic- integer technique, updating q^ME separately for each such maximum, and evaluating their respective likelihoods. The most likely phase sets are further refined by numerically maximizing their likelihood and are accepted as enlarged basis sets. This is a first approximate implementation of a general Bayesian theory of phase determination [Bricogne (1988). Acta Cryst. A44, 517-545]. A companion paper [Gilmore, Bricogne & Bannister (1990). Acta Cryst. A46, 297-308] describes successful applications to the ab initio phasing of small molecules, which demonstrate the viability of this new method and show that likelihood is far superior to any existing figure of merit in discriminating between correct and incorrect phases.