Quasar photometric redshifts beyond the spectroscopic coverage: Uncertainty models and redshift distributions in the Kilo-Degree Survey DR5

Abstract

Photometric redshifts (photo-$z$) and their distributions underpin cosmology with photometric quasars, tracers in angular clustering and cross-correlations. Progress requires trustworthy uncertainties, especially beyond the spectroscopic training set. We compare how machine-learning frameworks estimate quasar photo-$z$ uncertainties and reconstruct the redshift distribution $n(z)$ under controlled data-quality shifts. Using Kilo-Degree Survey DR5 photometry and DESI DR1 spectroscopic quasars, we train artificial neural networks (ANNs), Mixture Density Networks (MDNs) and Bayesian Neural Networks (BNNs) with Gaussian-mixture outputs, plus a self-organizing map (SOM) as a direct $n(z)$ estimator. We evaluate them on four subsets, with and without magnitude extrapolation and missing bands, through the negative log-likelihood, probability integral transform, point-estimate accuracy and the bias of binned $n(z)$ moments; degeneracies are sought by clustering the predicted PDFs. At least two mixture components are essential: a single Gaussian is miscalibrated and produces more catastrophic outliers than the ANN. No model performs best everywhere. On well-covered data the five-component MDN, three-component BNN and SOM reconstruct $n(z)$ almost perfectly; under faint extrapolation the BNN gives the best likelihoods, while in the hardest faint-plus-missing regime the single Gaussian becomes the best-calibrated. For point estimates and tomographic binning, uncertainty models outperform the ANN, while the SOM fails out-of-distribution. PDF clustering exposes distinct colour-redshift degeneracies likely to grow for fainter samples. The best model is thus regime- and application-dependent: multi-component MDNs or BNNs are needed for clean binning and are the only intrinsically calibrated choice on a well-covered golden sample, a first step towards a full comparison of quasar photo-$z$ pipelines.