Estimation of weights and validation: a marginal likelihood approach

The estimation of weights is quite an important aspect of the restrained refinement of macromolecular structures and related procedures such as the estimation of coordinate errors and structure validation using geometrical criteria. In principle, the method of maximum likelihood can be used for estimation of both atomic and weighting parameters. However, the low observation-to-parameter ratio in macromolecular refinement makes this kind of estimate of weighting parameters seriously biased; thus, the weighting parameters have traditionally been estimated separately from atomic parameters using a special technique, such as minimizing the free R factor. However, the variance of the latter estimate is large, as only a small portion of all data is used. In this work, an estimator of weights is proposed which is based on an approximation of a marginal likelihood function of the weighting parameters and which uses all the X-ray data. There is a known true value for the overall scaling coefficient for distance variances in restrained refinement and therefore the (maximum-likelihood) estimator for this coefficient may be used as a validation statistic.