How to obtain the redshift distribution from probabilistic redshift
  estimates
  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 $\texttt{chippr}$ 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.
