Fisher's information in maximum-likelihood macromolecular crystallographic refinement

Fisher's information is a statistical quantity related to maximum-likelihood theory. It is a matrix defined as the expected value of the squared gradient of minus the log-likelihood function. This matrix is positive semidefinite for any parameter value. Fisher's information is used in the quasi-Newton scoring method of minimization to calculate the shift vectors of model parameters. If the matrix is non-singular, the scoring-minimization step is always downhill. In this article, it is shown how the scoring method can be applied to macromolecular crystallographic refinement. It is also shown how the computational costs involved in calculation of the Fisher's matrix can be efficiently reduced. Speed is achieved by assuming a continuous distribution of reciprocal-lattice points. Matrix elements calculated with this method agree very well with those calculated analytically. The scoring algorithm has been implemented in the program REFMAC5 of the CCP4 suite. The Fisher's matrix is used in its sparse approximation. Tests indicate that the algorithm performs satisfactorily.