Photo-z Estimation: An Example of Nonparametric Conditional Density Estimation under Selection Bias

TL;DR

This paper introduces a nonparametric framework for estimating the full conditional density of galaxy redshifts from photometric data, addressing selection bias and combining multiple estimators, with applications to cosmology.

Contribution

It presents a novel framework for nonparametric conditional density estimation under selection bias, including methods to combine estimators for improved accuracy.

Findings

New estimators demonstrated on Sloan Data Sky Survey data

Effective handling of selection bias in high-dimensional density estimation

Applications to galaxy-galaxy lensing data

Abstract

Redshift is a key quantity for inferring cosmological model parameters. In photometric redshift estimation, cosmologists use the coarse data collected from the vast majority of galaxies to predict the redshift of individual galaxies. To properly quantify the uncertainty in the predictions, however, one needs to go beyond standard regression and instead estimate the full conditional density f(z|x) of a galaxy's redshift z given its photometric covariates x. The problem is further complicated by selection bias: usually only the rarest and brightest galaxies have known redshifts, and these galaxies have characteristics and measured covariates that do not necessarily match those of more numerous and dimmer galaxies of unknown redshift. Unfortunately, there is not much research on how to best estimate complex multivariate densities in such settings. Here we describe a general framework for properly constructing and assessing nonparametric conditional density estimators under selection bias, and for combining two or more estimators for optimal performance. We propose new improved photo-z estimators and illus- trate our methods on data from the Sloan Data Sky Survey and an application to galaxy-galaxy lensing. Although our main application is photo-z estimation, our methods are relevant to any high-dimensional regression setting with complicated asymmetric and multimodal distributions in the response variable.