  We describe a technique for simultaneously classifying and estimating the
redshift of quasars. It can separate quasars from stars in arbitrary redshift
ranges, estimate full posterior distribution functions for the redshift, and
naturally incorporate flux uncertainties, missing data, and multi-wavelength
photometry. We build models of quasars in flux-redshift space by applying the
extreme deconvolution technique to estimate the underlying density. By
integrating this density over redshift one can obtain quasar flux-densities in
different redshift ranges. This approach allows for efficient, consistent, and
fast classification and photometric redshift estimation. This is achieved by
combining the speed obtained by choosing simple analytical forms as the basis
of our density model with the flexibility of non-parametric models through the
use of many simple components with many parameters. We show that this technique
is competitive with the best photometric quasar classification
techniques---which are limited to fixed, broad redshift ranges and high
signal-to-noise ratio data---and with the best photometric redshift techniques
when applied to broadband optical data. We demonstrate that the inclusion of UV
and NIR data significantly improves photometric quasar--star separation and
essentially resolves all of the redshift degeneracies for quasars inherent to
the ugriz filter system, even when included data have a low signal-to-noise
ratio. For quasars spectroscopically confirmed by the SDSS 84 and 97 percent of
the objects with GALEX UV and UKIDSS NIR data have photometric redshifts within
0.1 and 0.3, respectively, of the spectroscopic redshift; this amounts to about
a factor of three improvement over ugriz-only photometric redshifts. Our code
to calculate quasar probabilities and redshift probability distributions is
publicly available.
