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I am now completely confused by the ML vs. LS discussion and hope that I am not the only person with this problem!
To clarify the issue, here is a simple clear-cut example of considerable interest to small-molecule crystallographers: what is the best way to determine the 'absolute configuration' using weak anomalous scattering (e.g. for an organic compound containing only C, H, N and O).
The conventional wisdom is that a so-called Flack parameter x is refined as one of the LEAST-SQUARES parameters, where Iobs(hkl) is fitted by Icalc(hkl) + x.Icalc(-h-k-l) with weights based on sigma(Iobs). Complex scattering factors are employed to calculate the intensity Icalc. This is similar to the way twins are refined.
When x is 0 (with a 'small' esd from the full-matrix LS) the absolute configuration is correct, when it is 1 (with a 'small' esd) it is wrong. If there are no errors in the model then only these two values of x are possible (racemic twinning, which would permit x to take any value in the range 0 to 1, is rare but not unknown; however in some case it can be ruled out by chemical evidence). Clearly the estimated esd of x is just as important as the actual value of x.
The following 'ML' approach was proposed in discussions at the Computing School in Siena (contributions from Rob Hooft, Simon Parsons and David Watkin are acknowledged, but I'm sure that there were others). We define the quantity Qobs = [Iobs(hkl)-Iobs(-h-k-l)]/[Iobs(hkl)+Iobs(-h-k-l)] and a corresponding Qcalc. This should cancel some systematic errors. The esd of Qobs can be calculated by standard methods from the known esds of Iobs(hkl) and Iobs(-h-k-l). Then we estimate the log likelihood (LL) of a particular value of x by -0.5[(Qobs-Qcalc)/esd(Qobs)]^2 summed over all reflections.
So my question is: should I estimate the value of x and its esd by integrating over -infinity < x < +infinity (in practice this range can be truncated to say -5 to +5) or should I integrate over 0 =< x =< 1 or should I use only the values of the LL at x = 0 and x = 1? Preliminary tests suggest that these methods give appreciably lower esds for x than the LS approach. If I only consider the LL for x=0 and x=1 I have a simple way of calculating the probability that x is 0 rather than 1 (i.e. the probability that the absolute configuration of the model is correct). Note that because of correlations between x and the atomic positional parameters in polar space groups it might be necessary to refine the structure to convergence for x=0 and x=1 to get the Icalc values for this.
I would appreciate clarification from the ML experts, with a view to putting it into my programs (which are still used by some small molecule crystallographers).
George
