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The next generation of cosmology experiments will be required to usephotometric redshifts rather than spectroscopic redshifts. Obtaining accurateand well-characterized photometric redshift distributions is therefore criticalfor Euclid, the Large Synoptic Survey Telescope and the Square Kilometre Array.However, determining accurate variance predictions alongside single pointestimates is crucial, as they can be used to optimize the sample of galaxiesfor the specific experiment (e.g. weak lensing, baryon acoustic oscillations,supernovae), trading off between completeness and reliability in the galaxysample. The various sources of uncertainty in measurements of the photometryand redshifts put a lower bound on the accuracy that any model can hope toachieve. The intrinsic uncertainty associated with estimates is oftennon-uniform and input-dependent, commonly known in statistics asheteroscedastic noise. However, existing approaches are susceptible to outliersand do not take into account variance induced by non-uniform data density andin most cases require manual tuning of many parameters. In this paper, wepresent a Bayesian machine learning approach that jointly optimizes the modelwith respect to both the predictive mean and variance we refer to as Gaussianprocesses for photometric redshifts (GPz). The predictive variance of the modeltakes into account both the variance due to data density and photometric noise.Using the SDSS DR12 data, we show that our approach substantially outperformsother machine learning methods for photo-z estimation and their associatedvariance, such as TPZ and ANNz2. We provide a Matlab and Python implementationsthat are available to download at https://github.com/OxfordML/GPz .

GPz: Non-stationary sparse Gaussian processes for heteroscedastic uncertainty estimation in photometric redshifts