  Extreme deconvolution (XD) of broad-band photometric data can both separate
stars from quasars and generate probability density functions for quasar
redshifts, while incorporating flux uncertainties and missing data.
Mid-infrared photometric colors are now widely used to identify hot dust
intrinsic to quasars, and the release of all-sky WISE data has led to a
dramatic increase in the number of IR-selected quasars. Using forced-photometry
on public WISE data at the locations of SDSS point sources, we incorporate this
all-sky data into the training of the XDQSOz models originally developed to
select quasars from optical photometry. The combination of WISE and SDSS
information is far more powerful than SDSS alone, particularly at $z>2$. The
use of SDSS$+$WISE photometry is comparable to the use of
SDSS$+$ultraviolet$+$near-IR data. We release a new public catalogue of
5,537,436 (total; 3,874,639 weighted by probability) potential quasars with
probability $P_{\textrm{QSO}} > 0.2$. The catalogue includes redshift
probabilities for all objects. We also release an updated version of the
publicly available set of codes to calculate quasar and redshift probabilities
for various combinations of data. Finally, we demonstrate that this method of
selecting quasars using WISE data is both more complete and efficient than
simple WISE color-cuts, especially at high redshift. Our fits verify that above
$z \sim 3$ WISE colors become bluer than the standard cuts applied to select
quasars. Currently, the analysis is limited to quasars with optical
counterparts, and thus cannot be used to find highly obscured quasars that WISE
color-cuts identify in significant numbers.
