Keywords: phases errors, likelihood function, ML-estimates, MLL-estimates, R-free.

The phases quality estimates derived from the maximization of the likelihood function (ML-estimates) were shown to be superior in comparison with other ones when dealing with phases calculated from an atomic model with independent coordinate errors [1] and were incorporated into wide-spread program SIGMAA. Nevertheless, it was found that ML-estimates have essential bias and overestimate the model quality when used for the atomic models refined in the reciprocal space [1, 2, 3].

The use of marginal likelihood function calculated in the framework of R-free-approach on the base of the test reflections only [4] allows one to obtain more reliable estimates (MML-estimates) both for unrefined and refined models [3, 5]. At the same time MML-estimates being derived on the base of a relatively small number of observations are prone to large statistical deviations from the mean. A simple smoothing procedure applied to structure factor distribution parameters allows one to reduce the statistical spread in the estimates significantly.

[1] Read, R. J. (1986). Acta Cryst. A42, 140-149.

[2] Lunin, V. Yu. & Urzhumtsev, A. G. (1984). Acta Cryst. A40, 265-277.

[3] Lunin, V. Yu. & Skovoroda, T. P. (1995). Acta Cryst. A51, 880-887.

[4] Brnger, A. T. (1992). Nature (London), 355, 472-474.

[5] Urzhumtsev, A. G., Skovoroda T. P. & Lunin V. Yu. (1996). J. Appl. Cryst. 29, 741-744.