Research Article: Estimating the difference between structure-factor amplitudes using multivariate Bayesian inference

Date Published: May 01, 2016

Publisher: International Union of Crystallography

Author(s): Gergely Katona, Maria-José Garcia-Bonete, Ida V. Lundholm.


A Bayesian model which uses a Markov chain Monte Carlo algorithm has been developed to estimate structure-factor amplitude differences.

Partial Text

There appears to be a strong dichotomy in the physical world between the observables and underlying wavefunctions (Dyson, 2007 ▸), and Bayesian models naturally lend an appropriate framework to discover the hidden parameters of the two-layered physical world. In X-ray crystallography the intensities of equivalent Bragg reflections are observed multiple times from which the directly unobservable structure-factor amplitudes can be estimated. Several X-ray crystallographic techniques exploit the fact that structure factors are dynamic and another (sub)structure can manifest itself as a difference in intensity observations. If the two sets of intensity observations are well separated in time or performed on different crystals there is a substantial risk that the systematic errors distort the difference amplitude estimates. To reduce the systematic errors between the observation sets, measurements can be taken from the same crystal and the intensities can be measured either simultaneously (González, 2003 ▸; Marinelli et al., 2015 ▸) or in rapidly alternating cycles (Lundholm et al., 2015 ▸; Westenhoff et al., 2010 ▸).

Two simulated intensity observation sets were generated based on the true value of structure-factor amplitudes F1true and F2true.

We have shown that a multivariate Bayesian model can provide more accurate structure-factor amplitude estimates from pairwise recorded diffraction intensities than univariate modelling. We also demonstrated that this multivariate model can be efficiently evaluated by an MCMC algorithm. We anticipate that the accuracy gains will lead to improved phasing results and more detailed difference electron-density maps in time-resolved pump–probe diffraction experiments.




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