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.