  We have combined a survey at 24 microns to 0.3 mJy with the Multiband Imaging
Photometer (MIPS) on the Spitzer Space Telescope, a 20 cm A-array VLA survey
covering 0.5 deg^2, and an existing 172 ks Chandra X-Ray Observatory exposure
to investigate the nature of optically faint radio and X-ray sources in the
NOAO Deep Wide-Field Survey (NDWFS) in Bootes. We find little overlap between
the radio and infrared selected populations: only 9% of the infrared sources
are detected in the radio and only 33% of the radio sources are detected in the
infrared. Thirty-six (10%) of the 377 compact radio sources lack optical
counterparts in the NDWFS Bw, R, & I images. We refer to these objects as
optically invisible radio sources (OIRS). Only four of the thirty-one OIRSs
observed with MIPS are detected at 24 microns. Comparisons of the radio and
infrared properties of the OIRSs with various galaxy spectral energy
distributions demonstrate that most of these sources are powered by AGN rather
than starbursts. Similarly, eleven X-ray sources observed by both MIPS and the
VLA are classified as optically invisible X-ray sources (OIXS). None are
detected at 24 microns or 20 cm. All seven OIXSs detected in Chandra's 0.5-2
keV band have infrared to X-ray flux ratios consistent with their luminosity
being dominated by an unobscured AGN. From these results we conclude that both
the optically invisible radio and X-ray source populations are primarily AGN,
relatively unaffected by dust and most likely at z > 1. No OIRSs are detected
in X-ray emission and no OIXSs are detected at 20 cm. However, given the wide
range in radio and X-ray properties of known AGN and the size of our samples,
this lack of overlap does not necessarily imply AGN source populations.
