We discuss the implementation of Bayesian inversion methods in order to recover the properties of the intergalactic medium from observations of the neutral hydrogen Lyman α absorptions observed in the spectra of high-redshift quasars (the so-called Lyman α forest). We use two complementary schemes: (i) a constrained Gaussian random field linear approach, and (ii) a more general non-linear explicit Bayesian deconvolution method, which offers in particular the possibility to constrain the parameters of the equation of state for the gas. The interpolation ability of the first approach is shown to be equivalent to the second one in the limit of negligible measurement errors, low-resolution spectra and null mean prior. While relying on prior assumption for the two-point correlation functions, we show how to recover, at least qualitatively, the three-dimensional topology of the large-scale structures in redshift space by inverting a suitable network of adjacent, low-resolution lines of sight. The methods are tested on regular bundles of lines of sight using N-body simulations specially designed to tackle this problem. We also discuss the inversion of single lines of sight observed at high spectral resolution. Our preliminary investigations suggest that the explicit Bayesian method can be used to derive quantitative information on the physical state of the gas when the effects of redshift distortion are negligible. The information in the spectra remains degenerate with respect to two parameters (the temperature scale factor and the polytropic index) describing the equation of state of the gas. Redshift distortion is considered by simultaneous constrained reconstruction of the velocity and the density field in real space, while assuming statistical correlation between the two fields. The method seems to work well in the strong prior régime where peculiar velocities are assumed to be the most likely realization in the density field. Finally, we investigate the effect of line-of-sight separation and number of lines of sight. Our analyses suggest that multiple low-resolution lines of sight could be used to improve the most likely velocity reconstruction on a high-resolution line of sight.