How to obtain the redshift distribution from probabilistic redshift estimates
Abstract
A trustworthy estimate of the redshift distribution n(z) is crucial for using weak gravitational lensing and large-scale structure of galaxy catalogs to study cosmology. Spectroscopic redshifts for the dim and numerous galaxies of next-generation weak-lensing surveys are expected to be unavailable, making photometric redshift (photo-z) probability density functions (PDFs) the next-best alternative for comprehensively encapsulating the nontrivial systematics affecting photo-z point estimation. The established stacked estimator of n(z) avoids reducing photo-z PDFs to point estimates but yields a systematically biased estimate of n(z) that worsens with decreasing signal-to-noise, the very regime where photo-z PDFs are most necessary. We introduce Cosmological Hierarchical Inference with Probabilistic Photometric Redshifts (CHIPPR), a statistically rigorous probabilistic graphical model of redshift-dependent photometry, which correctly propagates the redshift uncertainty information beyond the best-fit estimator of n(z) produced by traditional procedures and is provably the only self-consistent way to recover n(z) from photo-z PDFs. We present the chippr prototype code, noting that the mathematically justifiable approach incurs computational expense. The CHIPPR approach is applicable to any one-point statistic of any random variable, provided the prior probability density used to produce the posteriors is explicitly known; if the prior is implicit, as may be the case for popular photo-z techniques, then the resulting posterior PDFs cannot be used for scientific inference. We therefore recommend that the photo-z community focus on developing methodologies that enable the recovery of photo-z likelihoods with support over all redshifts, either directly or via a known prior probability density.