We present the X-ray spectroscopic study of the Compton-thick (CT) active galactic nuclei (AGN) population within the $textit{Chandra}$ Deep Field South (CDF-S) by using the deepest X-ray observation to date, the $textit{Chandra}$ 7 Ms observation of the CDF-S. We combined an opimized version of our automated selection technique and a Bayesian Monte Carlo Markov Chains (MCMC) spectral fitting procedure, to develop a method to pinpoint and then characterize candidate CT AGN as less model dependent and/or data-quality dependent as possible. To obtain reliable automated spectral fits, we only considered the sources detected in the hard (2-8 keV) band from the CDF-S 2 Ms catalog with either spectroscopic or photometric redshifts available for 259 sources. Instead of using our spectral analysis to decide if an AGN is CT, we derived the posterior probability for the column density, and then we used it to assign a probability of a source being CT. We also tested how the model-dependence of the spectral analysis, and the spectral data quality, could affect our results by using simulations. We finally derived the number density of CT AGN by taking into account the probabilities of our sources being CT and the results from the simulations. Our results are in agreement with X-ray background synthesis models, which postulate a moderate fraction (25%) of CT objects among the obscured AGN population.
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