The method of using the ratio of gradients doesn't make
sense in a maximum likelihood context,
assuming that by "a maximum likelihood context" you mean refinement
using a maximum-likelihood (ML) criterion as X-ray term (or, more
generally, I would call it experimental data term, as it can be
neutron too, for instance), I find the whole statement above as a
little bit strange since it mixes different and absolutely not
related things: type of crystallographic data term and a method of
relative scale (weight) determination between it and the other term
(restraints).
I don't see how the choice of crystallographic data term (LS, ML,
real-space or any other) is related to the method of this scale
determination.
This shouldn't be a surprise - in short, the errors are used as weights in LS and ML optimization targets, the latter just uses a different form for the errors that estimates all the model and unmeasured uncertainties (like phase error).� So if the data is poorly predicted by a model, the ML target is broader/flatter (as are the gradients!), while good/complete models will yield a sharper ML target.� So the likelihood target is naturally weighted, in a sense.� This doesn't happen with least squares (unless the weights are not the inverse variances, which seems to be what the MLMF paper you mentioned is doing?).
The likelihood function can then be plugged in to Bayes' law - if the model and data error terms are all accounted for, no other weighting should be necessary.� This is discussed in Airlie McCoy's excellent review (
http://dx.doi.org/10.1107/S0907444904016038) - see sections 4.4 and 4.6, and the derivation is also in
http://dx.doi.org/10.1107/S0907444911039060
�