This paper presents a Bayesian model selection approach to analyze X-ray spectra of active galactic nuclei (AGN) in the Chandra Deep Field South (CDFS). The study aims to understand the obscuring material around supermassive black holes (SMBHs) and its impact on AGN spectra. The authors develop a Bayesian framework to compare different X-ray spectral models and estimate parameters, taking into account uncertainties from Poisson noise and redshift measurements. They apply this methodology to 350 AGN in the CDFS, identifying four key components in their X-ray spectra: an intrinsic power law, a cold obscurer with photo-electric absorption, Compton scattering, and Fe-K fluorescence; an unabsorbed power law associated with Thomson scattering; and Compton reflection. The analysis shows that simpler models are ruled out with strong evidence, and that ignoring Thomson scattering leads to underestimation of the obscuring column density. The geometry of the obscuring material is found to be intermediate between a closed (sphere) and open (blob) torus. The study confirms the presence of a soft component possibly linked to Thomson scattering in the torus opening angle and finds evidence for a density gradient or reflection off the accretion disk. The authors release a catalog of AGN in the CDFS with estimated parameters such as accretion luminosity and obscuring column density. The methodology is able to draw strong inferences on the torus geometry despite low-count spectra, and confirms the presence of a soft component. The study highlights the importance of X-ray observations in constraining AGN properties and their host galaxies. The results have implications for understanding AGN evolution and their role in galaxy formation. The paper also discusses the use of Bayesian methods for model comparison and parameter estimation in X-ray spectral analysis, and presents a framework for future studies. The authors conclude that their approach provides a robust method for analyzing AGN X-ray spectra and understanding the obscuring material around SMBHs.This paper presents a Bayesian model selection approach to analyze X-ray spectra of active galactic nuclei (AGN) in the Chandra Deep Field South (CDFS). The study aims to understand the obscuring material around supermassive black holes (SMBHs) and its impact on AGN spectra. The authors develop a Bayesian framework to compare different X-ray spectral models and estimate parameters, taking into account uncertainties from Poisson noise and redshift measurements. They apply this methodology to 350 AGN in the CDFS, identifying four key components in their X-ray spectra: an intrinsic power law, a cold obscurer with photo-electric absorption, Compton scattering, and Fe-K fluorescence; an unabsorbed power law associated with Thomson scattering; and Compton reflection. The analysis shows that simpler models are ruled out with strong evidence, and that ignoring Thomson scattering leads to underestimation of the obscuring column density. The geometry of the obscuring material is found to be intermediate between a closed (sphere) and open (blob) torus. The study confirms the presence of a soft component possibly linked to Thomson scattering in the torus opening angle and finds evidence for a density gradient or reflection off the accretion disk. The authors release a catalog of AGN in the CDFS with estimated parameters such as accretion luminosity and obscuring column density. The methodology is able to draw strong inferences on the torus geometry despite low-count spectra, and confirms the presence of a soft component. The study highlights the importance of X-ray observations in constraining AGN properties and their host galaxies. The results have implications for understanding AGN evolution and their role in galaxy formation. The paper also discusses the use of Bayesian methods for model comparison and parameter estimation in X-ray spectral analysis, and presents a framework for future studies. The authors conclude that their approach provides a robust method for analyzing AGN X-ray spectra and understanding the obscuring material around SMBHs.