Hierarchical representation of the BORG inference framework for the analysis of Ly-α forest data. Primordial fluctuations δ_ic encoded in a a set of Fourier modes at z ≈ 1000 are obtained from the prior P(δ_ic|Ω), where Ω represents the cosmological parameters. These initial conditions are evolved to z = 2.5 using PPT, which provides the optical depth τ_PPT and the evolved density, δ_PPT. The optical depth is then used to generate quasar spectra based on the fluctuating Gunn-Peterson approximation (FGPA). F_obs indicates the data. Purple boxes indicate deterministic transition while green boxes are probability distributions. Iterating this procedure results in a Monte Carlo Markov chain that explores the joint posterior distribution of the three-dimensional matter distribution underlying Ly-α forest observations.