All-purpose, all-sky photometric redshifts for the Legacy Imaging
  Surveys Data Release 8
  In this paper we present photometric redshift (photo-$z$) estimates for the
Dark Energy Spectroscopic Instrument (DESI) Legacy Imaging Surveys, currently
the most sensitive optical survey covering the majority of the extra-galactic
sky. Our photo-$z$ methodology is based on a machine-learning approach, using
sparse Gaussian processes augmented with Gaussian mixture models (GMMs) that
allow regions of parameter space to be identified and trained separately in a
purely data-driven way. The same GMMs are also used to calculate cost-sensitive
learning weights that mitigate biases in the spectroscopic training sample. By
design, this approach aims to produce reliable and unbiased predictions for all
parts of the parameter space present in wide area surveys. Compared to previous
literature estimates using the same underlying photometry, our photo-$z$s are
significantly less biased and more accurate at $z > 1$, with negligible loss in
precision or reliability for resolved galaxies at $z < 1$. Our photo-$z$
estimates offer accurate predictions for rare high-value populations within the
parent sample, including optically selected quasars at the highest redshifts
($z > 6$), as well as X-ray or radio continuum selected populations across a
broad range of flux (densities) and redshift. Deriving photo-$z$ estimates for
the full Legacy Imaging Surveys Data Release 8, the catalogues provided in this
work offer photo-$z$ estimates predicted to be high quality for
$\gtrsim9\times10^{8}$ galaxies over $\sim 19\,400\,\text{deg}^{2}$ and
spanning $0 < z \lesssim 7$, offering one of the most extensive samples of
redshift estimates ever produced.
