  We present a novel Bayesian method for the joint reconstruction of
cosmological matter density fields, peculiar velocities and power-spectra in
the quasi-nonlinear regime. We study its applicability to the Ly-alpha forest
based on multiple quasar absorption spectra. Our approach to this problem
includes a multiscale, nonlinear, two-step scheme since the statistics
describing the matter distribution depends on scale, being strongly
non-Gaussian on small scales (< 0.1 h^{-1} Mpc) and closely lognormal on scales
>~10 h^{-1} Mpc. The first step consists on performing 1D highly resolved
matter density reconstructions along the line-of-sight towards z~2-3 quasars
based on an arbitrary non-Gaussian univariate model for matter statistics. The
second step consists on Gibbs-sampling based on conditional PDFs. The matter
density field is sampled in real space with Hamiltonian-sampling using the
Poisson/Gamma-lognormal model, while redshift distortions are corrected with
linear Lagrangian perturbation theory. The power-spectrum of the lognormal
transformed variable which is Gaussian distributed (and thus close to the
linear regime) can consistently be sampled with the inverse Gamma distribution
function. We test our method through numerical N-body simulations with a
computational volume large enough (> 1 h^{-3} Gpc^3) to show that the linear
power-spectra are nicely recovered over scales larger than >~20 h^{-1} Mpc,
i.e. the relevant range where features imprinted by the baryon-acoustics
oscillations (BAOs) appear.
