  We combine data from two all-sky surveys in order to study the connection
between the infrared and hard X-ray (>10keV) properties for local active
galactic nuclei (AGN). The Swift/Burst Alert Telescope all-sky survey provides
an unbiased, flux-limited selection of hard X-ray detected AGN.
Cross-correlating the 22-month hard X-ray survey with the AKARI all-sky survey,
we studied 158 AGN detected by the AKARI instruments. We find a strong
correlation for most AGN between the infrared (9, 18, and 90 micron) and hard
X-ray (14-195 keV) luminosities, and quantify the correlation for various
subsamples of AGN. Partial correlation analysis confirms the intrinsic
correlation after removing the redshift contribution. The correlation for radio
galaxies has a slope and normalization identical to that for Seyfert 1s,
implying similar hard X-ray/infrared emission processes in both. In contrast,
Compton-thick sources show a large deficit in the hard X-ray band, because high
gas column densities diminish even their hard X-ray luminosities. We propose
two photometric diagnostics for source classification: one is an X-ray
luminosity vs. infrared color diagram, in which type 1 radio-loud AGN are well
isolated from the others in the sample. The other uses the X-ray vs. infrared
color as a useful redshift-independent indicator for identifying Compton-thick
AGN. Importantly, Compton-thick AGN and starburst galaxies in composite systems
can also be differentiated in this plane based upon their hard X-ray fluxes and
dust temperatures. This diagram may be useful as a new indicator to classify
objects in new and upcoming surveys such as WISE and NuSTAR.
