  A complete census of Active Galactic Nuclei (AGN) is a prerequisite for
understanding the growth of supermassive black holes across cosmic time. A
significant challenge toward this goal is the whereabouts of heavily obscured
AGN that remain uncertain. This paper sets new constraints on the demographics
of this population by developing a methodology that combines X-ray spectral
information with priors derived from multiwavelength observations. We select
X-ray AGN in the Chandra COSMOS Legacy survey and fit their $2.2-500\mu m$
spectral energy distributions with galaxy and AGN templates to determine the
mid-infrared ($6\mu m$) luminosity of the AGN component. Empirical correlations
between X-ray and $6\mu m$ luminosities are then adopted to infer the intrinsic
accretion luminosity at X-rays for individual AGN. This is used as prior
information in our Bayesian X-ray spectral analysis to estimate physical
properties, such as line-of-sight obscuration. Our approach breaks the
degeneracies between accretion luminosity and obscuration that affect X-ray
spectral analysis, particularly for the most heavily obscured (Compton-Thick)
AGN with low photon counts X-ray spectra. The X-ray spectral results are then
combined with the selection function of the Chandra COSMOS Legacy survey to
derive the AGN space density and a Compton-Thick fraction of
$21.0^{+16.1}_{-9.9}\%$ at redshifts $z<0.5$. At higher redshift, our analysis
suggests upper limits to the Compton-Thick AGN fraction of $\le 40\%$. These
estimates are at the low end of the range of values determined in the
literature and underline the importance of multiwavelength approaches for
tackling the challenge of heavily obscured AGN demographics.
