  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.
