  Heavily obscured active galactic nuclei (AGNs) play an important role in
contributing to the cosmic X-ray background (CXRB). However, the AGNs found in
deep X-ray surveys are often too weak to allow direct measurement of the column
density of obscuring matter. One method adopted in recent years to identify
heavily obscured, Compton-thick AGNs under such circumstances is to use the
observed mid-infrared to X-ray luminosity ratio as a proxy for the column
density. This is based on the supposition that the amount of energy lost by the
illuminating X-ray continuum to the obscuring matter and reprocessed into
infrared emission is directly related to the column density and that the proxy
is not sensitive to other physical parameters of the system (aside from
contamination by dust emission from, for example, star-forming regions). Using
Monte Carlo simulations, we find that the energy losses experienced by the
illuminating X-ray continuum in the obscuring matter are far more sensitive to
the shape of the X-ray continuum and to the covering factor of the X-ray
reprocessor than they are to the column density of the material. Specifically
we find that it is possible for the infrared to X-ray luminosity ratio for a
Compton-thin source to be just as large as that for a Compton-thick source even
without any contamination from dust. Since the intrinsic X-ray continuum and
covering factor of the reprocessor are poorly constrained from deep X-ray
survey data, we conclude that the mid-infrared to X-ray luminosity ratio is not
a reliable proxy for the column density of obscuring matter in AGNs even when
there is no other contribution to the mid-infrared luminosity aside from X-ray
reprocessing. This conclusion is independent of the geometry of the obscuring
matter.
