Reliability of maximum likelihood-based figures of merit

Various schemes for determining the maximum likelihood-based figures of merit for phases of structure factors have been considered. It is shown that the use of the likelihood function of all the available structure factors provides the adequate estimates of the accuracy of phases calculated for the atomic models with independent errors in the coordinates, but, at the same time, systematically overestimates the figures of merit for models preliminarily refined in the reciprocal space. It is shown that the use of the marginal likelihood function calculated from the control set of reflections allows the elimination of the systematic bias estimates. A method for reducing the statistical dispersion of the estimates based on a small number of control reflections is suggested.