*J. Appl. Cryst.* (2013). **46**, 404-414 [ doi:10.1107/S002188981300109X ]

**Abstract:** The interatomic distance distribution, *P*(*r*), is a valuable tool for evaluating the structure of a molecule in solution and represents the maximum structural information that can be derived from solution scattering data without further assumptions. Most current instrumentation for scattering experiments (typically CCD detectors) generates a finely pixelated two-dimensional image. In continuation of the standard practice with earlier one-dimensional detectors, these images are typically reduced to a one-dimensional profile of scattering intensities, *I*(*q*), by circular averaging of the two-dimensional image. Indirect Fourier transformation methods are then used to reconstruct *P*(*r*) from *I*(*q*). Substantial advantages in data analysis, however, could be achieved by directly estimating the *P*(*r*) curve from the two-dimensional images. This article describes a Bayesian framework, using a Markov chain Monte Carlo method, for estimating the parameters of the indirect transform, and thus *P*(*r*), directly from the two-dimensional images. Using simulated detector images, it is demonstrated that this method yields *P*(*r*) curves nearly identical to the reference *P*(*r*). Furthermore, an approach for evaluating spatially correlated errors (such as those that arise from a detector point spread function) is evaluated. Accounting for these errors further improves the precision of the *P*(*r*) estimation. Experimental scattering data, where no ground truth reference *P*(*r*) is available, are used to demonstrate that this method yields a scattering and detector model that more closely reflects the two-dimensional data, as judged by smaller residuals in cross-validation, than *P*(*r*) obtained by indirect transformation of a one-dimensional profile. Finally, the method allows concurrent estimation of the beam center and *D*_{max}, the longest interatomic distance in *P*(*r*), as part of the Bayesian Markov chain Monte Carlo method, reducing experimental effort and providing a well defined protocol for these parameters while also allowing estimation of the covariance among all parameters. This method provides parameter estimates of greater precision from the experimental data. The observed improvement in precision for the traditionally problematic *D*_{max} is particularly noticeable.

**Keywords: structure analysis; small-angle X-ray scattering; small-angle neutron scattering; Bayesian inference; Markov chain Monte Carlo methods.**

| |

To open or display or play some files, you may need to set your browser up to use the appropriate software. See the full list of file types for an explanation of the different file types and their related mime types and, where available links to sites from where the appropriate software may be obtained.

The download button will force most browsers to prompt for a file name to store the data on your hard disk.

Where possible, images are represented by thumbnails.

Copyright © International Union of Crystallography

*IUCr Webmaster*