Section 15.2.9. Maximum-likelihood structure refinement

In the past, conventional structure refinement was based on a least-squares target, which would be justified if the observed and calculated structure-factor amplitudes were related by a Gaussian probability distribution. Unfortunately, the relationship between and is not Gaussian, and the distribution for is not even centred on . Because of this, it was suggested (Read, 1990; Bricogne, 1991) that a maximum-likelihood target should be used instead, and that it should be based on probability distributions such as those described above.

Three implementations of maximum-likelihood structure refinement have now been reported (Pannu & Read, 1996; Murshudov et al., 1997; Bricogne & Irwin, 1996). As expected, there is a decrease in refinement bias, as the calculated structure-factor amplitudes will not be forced to be equal to the observed amplitudes. Maximum-likelihood targets have been shown to work much better than least-squares targets, particularly when the starting models are poor.

Prior phase information can also be incorporated into a maximum-likelihood target (Pannu et al., 1998). Tests show that even weak phase information can have a dramatic effect on the success of refinement, and that the amount of overfitting is even further reduced (Pannu et al., 1998).