"We address in this proposal the inverse problem for the reconstruction of the properties of the intergalactic medium based on the measured neutral hydrogen Lyman-α absorption observed in the spectra of high redshift quasars. We suggest the application of ARGO, a Bayesian sampling solver that has been developed by the applicant during his PhD thesis for the reconstruction of the large-scale structure and its properties based on galaxy redshift-surveys. This will permit for the first time to have a self-consistent reconstruction of all the relevant properties of the intergalactic medium. Some further developments are required in order to extent this work to the reconstruction of the intergalactic medium that should be implemented during this training. However, the numerical technology developed by the applicant A novel Krylov-space based inverse scheme included in ARGO allows for efficient reconstruction steps as is required for sampling purposes even in huge dimensional spaces. The cosmological information extracted from this study, such as the cosmological parameters is expected to be qualitatively better than in previous works in this field. This project should have major contributions in the understanding of galaxy and structure formation"