A Unified Framework for Constructing, Tuning and Assessing Photometric Redshift Density Estimates in a Selection Bias Setting

Abstract

Photometric redshift estimation is an indispensable tool of precision cosmology. One problem that plagues the use of this tool in the era of large-scale sky surveys is that the bright galaxies that are selected for spectroscopic observation do not have properties that match those of (far more numerous) dimmer galaxies; thus, ill-designed empirical methods that produce accurate and precise redshift estimates for the former generally will not produce good estimates for the latter. In this paper, we provide a principled framework for generating conditional density estimates (i.e. photometric redshift PDFs) that takes into account selection bias and the covariate shift that this bias induces. We base our approach on the assumption that the probability that astronomers label a galaxy (i.e. determine its spectroscopic redshift) depends only on its measured (photometric and perhaps other) properties x and not on its true redshift. With this assumption, we can explicitly write down risk functions that allow us to both tune and compare methods for estimating importance weights (i.e. the ratio of densities of unlabeled and labeled galaxies for different values of x) and conditional densities. We also provide a method for combining multiple conditional density estimates for the same galaxy into a single estimate with better properties. We apply our risk functions to an analysis of approximately one million galaxies, mostly observed by SDSS, and demonstrate through multiple diagnostic tests that our method achieves good conditional density estimates for the unlabeled galaxies.

11 pages; accepted by MNRAS

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• Assessing the photometric redshift precision of the S-PLUS survey: the Stripe-82 as a test-case
• Flexible distribution-free conditional predictive bands using density estimators
• SKA Science Data Challenge 2: analysis and results
• A unified framework for dataset shift diagnostics
• CD-split and HPD-split: efficient conformal regions in high dimensions
• pop-cosmos: Scaleable inference of galaxy properties and redshifts with a data-driven population model
• Diagnostics for Conditional Density Models and Bayesian Inference Algorithms
• The sensitivity of GPz estimates of photo-z posterior PDFs to realistically complex training set imperfections
• The Quasar Catalogue for S-PLUS DR4 (QuCatS) and the estimation of photometric redshifts
• Improved Weak Lensing Photometric Redshift Calibration via StratLearn and Hierarchical Modeling
• Mapping Variations of Redshift Distributions with Probability Integral Transforms
• Quantifying Epistemic Uncertainty in Deep Learning