Statistical approach to the phase problem.

The minimal function and its minimal principle employed in the traditional Shake-and-Bake algorithm rely on the probabilistic estimates of the cosines of the structure invariants. In this paper, a novel statistical approach to the phase problem, which utilizes statistical properties of the structure invariants, is proposed. The statistical maximal function and its maximal principle are formulated, and the corresponding statistical Shake-and-Bake algorithm and its associated statistical parameter-shift procedure are proposed and tested. The test results show that the statistical approach to the phase problem is a simple, reliable, less computationally intensive and more efficient procedure for phase determination in X-ray crystallography.