Top panel: posterior predictive flux for a spectrum with noise σ = 0.03. The posterior predicted flux is computed from the ensemble mean density field, shown in the lower panel (orange line). The blue shaded region indicates the 1-σ region, corresponding to the standard deviation of the noise in this line of sight. These tests check whether the data model can accurately account for the observations. Any significant mismatch would immediately indicate a breakdown of the applicability of the data model or error of the inference framework. Our method recovers the transmitted flux fraction correctly, confirming that the data model can accurately account for the observations. Bottom panel: comparison of the inferred ensemble mean density field along the line of sight to the ground truth. It can be seen that high-density regions yield a suppression of transmitted flux, while under-dense regions transmit the quasar signal, in agreement with the FGPA model. The density field has been smoothed with a Gaussian smoothing kernel of σ = 0.5 h⁻¹ Mpc to simulate the difference between dark matter and gas density fields (Peirani et al. 2014). This smoothing kernel is applied to the dark matter density field before generating the mock data.