Mid-Infrared Selection of Active Galactic Nuclei with the Wide-Field
  Infrared Survey Explorer. II. Properties of WISE-Selected Active Galactic
  Nuclei in the NDWFS Bo\"otes Field
  Stern et al.(2012) presented a study of WISE selection of AGN in the 2 deg^2
COSMOS field, finding that a simple criterion W1-W2>=0.8 provides a highly
reliable and complete AGN sample for W2<15.05, where the W1 and W2 passbands
are centered at 3.4 and 4.6 microns, respectively. Here we extend this study
using the larger 9 deg^2 NOAO Deep Wide-Field Survey Bootes field which also
has considerably deeper WISE observations than the COSMOS field, and find that
this simple color-cut significantly loses reliability at fainter fluxes. We
define a modified selection criterion combining the W1-W2 color and the W2
magnitude to provide highly reliable or highly complete AGN samples for fainter
WISE sources. In particular, we define a color-magnitude cut that finds 130+/-4
deg^-2 AGN candidates for W2<17.11 with 90% reliability. Using the extensive UV
through mid-IR broad-band photometry available in this field, we study the
spectral energy distributions of WISE AGN candidates. As expected, the WISE AGN
selection is biased towards objects where the AGN dominates the bolometric
luminosity output, and that it can identify highly obscured AGN. We study the
distribution of reddening in the AGN sample and discuss a formalism to account
for sample incompleteness based on the step-wise maximum-likelihood method of
Efstathiou et al.(1988). The resulting dust obscuration distributions depend
strongly on AGN luminosity, consistent with the trend expected for a Simpson
(2005) receding torus. At L_AGN~3x10^44 erg/s, 29+/-7% of AGN are observed as
Type 1, while at ~4x10^45 erg/s the fraction is 64+/-13%. The distribution of
obscuration values suggests that dust in the torus is present as both a diffuse
medium and in optically thick clouds.
