Slices through ground truth initial (left upper panel), final density field (left lower panel), inferred ensemble mean initial (middle upper panel) and ensemble mean final (middle-lower panel) density field computed from 12600 MCMC samples. Comparison between these panels shows that the method recovers the structure of the true density field with high accuracy. We note that the algorithm infers the correct amplitudes of the density field. Right panels: standard deviations of inferred amplitudes of initial (upper right panel) and final density fields (lower right panel). The standard deviation of the final density field shows lower values at the position of the lines of sight. We note that the uncertainty of δ^f presents a structure that correlates with the density field. Particularly, the variance is higher in high-density regions due to the saturation of the absorbed flux. In contrast, the standard deviation of the initial conditions are homogeneous and show no correlation with the initial density field due to the propagation of information from the final to the initial density field via the dynamical model.