The notion of recentring and the maximum-entropy criterion

Bricogne, G.

doi:10.1107/97809553602060000690

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 16.2, p. 348 | 1 | 2 |

Section 16.2.3.3. The notion of recentring and the maximum-entropy criterion

G. Bricogne^a ^*

^aLaboratory of Molecular Biology, Medical Research Council, Cambridge CB2 2QH, England
Correspondence e-mail: gb10@mrc-lmb.cam.ac.uk

16.2.3.3. The notion of recentring and the maximum-entropy criterion

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These limitations can be overcome by recognizing that, if the locus $[{\cal T}]$ (a high-dimensional torus) defined by the large structure-factor amplitudes to be used in the conditioning data is too extended in structure-factor space for a single asymptotic expansion of $[{\cal P}({\bf F})]$ to be accurate everywhere on it, then $[{\cal T}]$ should be broken up into sub-regions, and different local approximations to $[{\cal P}({\bf F})]$ should be constructed in each of them. Each of these sub-regions will consist of a `patch' of $[{\cal T}]$ surrounding a point $[{\bf F^{*} \neq 0}]$ located on $[{\cal T}]$ . Such a point $[{\bf F^{*}}]$ is obtained by assigning `trial' phase values to the known moduli, but these trial values do not necessarily have to be viewed as `serious' assumptions concerning the true values of the phases: rather, they should be thought of as pointing to a patch of $[{\cal T}]$ and to a specialized asymptotic expansion of $[{\cal P}({\bf F})]$ designed to be the most accurate approximation possible to $[{\cal P}({\bf F})]$ on that patch. With a sufficiently rich collection of such constructs, $[{\cal P}({\bf F})]$ can be accurately calculated anywhere on $[{\cal T}]$ .

These considerations lead to the notion of recentring . Recentring the usual Gram–Charlier or Edgeworth asymptotic expansion for $[{\cal P}({\bf F})]$ away from $[{\bf F} = {\bf 0}]$ , by making trial phase assignments that define a point $[{\bf F^{*}}]$ on $[{\cal T}]$ , is equivalent to using a non-uniform prior distribution of atoms q(x), reproducing the individual components of $[{\bf F^{*}}]$ among its Fourier coefficients. The latter constraint leaves q(x) highly indeterminate, but Jaynes' argument given in Section 16.2.2.3 shows that there is a uniquely defined `best' choice for it: it is that distribution $[q^{\rm ME}({\bf x})]$ having maximum entropy relative to a uniform prior prejudice m(x), and having the corresponding values $[{\bf U^{*}}]$ of the unitary structure factors for its Fourier coefficients. This distribution has the unique property that it rules out as few random structures as possible on the basis of the limited information available in $[{\bf F^{*}}]$ .

In terms of the statistical mechanical language used in Section 16.2.1, the trial structure-factor values $[{\bf F^{*}}]$ used as constraints would be the macroscopic quantities that can be controlled externally; while the 3N atomic coordinates would be the internal degrees of freedom of the system, whose entropy should be a maximum under these macroscopic constraints.

References

International Tables for Crystallography (2006). Vol. F. ch. 16.2, p. 348