Fig. 2: Estimation of scales and structure factor amplitudes from simulated data.
From: A unifying Bayesian framework for merging X-ray diffraction data
a A probabilistic graphical model summarizes our basic statistical formalism: Careless calculates a probabilistic scale, Σ, as a function of the recorded metadata, M, and learned parameters, θ. Observed intensities, Ih,i, for Miller index h and image i are modeled as the product of the scale and the square of the structure factor amplitude Fh, that is, as \({{{\Sigma }}}_{h,i}\cdot {F}_{h}^{2}\). b The global scale function that maps the recorded metadata to the probabilistic scale, Σ, takes the form of a multilayer perceptron. c–e Inference of scales and structure factors from simulated data comprising 10 draws from the ground truth model. (c) Input parameters for the simulated data were chosen to recapitulate the non-linear scales observed in diffraction data. d Noisy observations were generated from these input parameters, which reflects the measurement errors in a diffraction experiment (ten per structure factor). Shown are simulated measured values. Error bars indicate 95% confidence intervals. e This statistical model allows for joint, rather than sequential, inference of the posterior distributions of structure factor amplitudes and scales, and therefore of implied intensity (bottom panel). The violin plots in the top panel show the posterior probability with whiskers indicating the extrema of 10,000 samples drawn from the posterior distribution of the inferred F. Shaded bands indicate 95% confidence intervals around the posterior means. The posterior mean of the scale function is indicated as a dashed line (middle panel), and the posterior mean for the reflection intensities are shown as circles (bottom panel). For this toy example, the inferred values in (e) can be compared to the known ground truth in (c).
