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A trustworthy estimate of the redshift distribution $n(z)$ is crucial forusing weak gravitational lensing and large-scale structure of galaxy catalogsto study cosmology. Spectroscopic redshifts for the dim and numerous galaxiesof next-generation weak-lensing surveys are expected to be unavailable, makingphotometric redshift (photo-$z$) probability density functions (PDFs) thenext-best alternative for comprehensively encapsulating the nontrivialsystematics affecting photo-$z$ point estimation. The established stackedestimator of $n(z)$ avoids reducing photo-$z$ PDFs to point estimates butyields a systematically biased estimate of $n(z)$ that worsens with decreasingsignal-to-noise, the very regime where photo-$z$ PDFs are most necessary. Weintroduce Cosmological Hierarchical Inference with Probabilistic PhotometricRedshifts (CHIPPR), a statistically rigorous probabilistic graphical model ofredshift-dependent photometry, which correctly propagates the redshiftuncertainty information beyond the best-fit estimator of $n(z)$ produced bytraditional 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 computationalexpense. The CHIPPR approach is applicable to any one-point statistic of anyrandom variable, provided the prior probability density used to produce theposteriors is explicitly known; if the prior is implicit, as may be the casefor popular photo-$z$ techniques, then the resulting posterior PDFs cannot beused for scientific inference. We therefore recommend that the photo-$z$community focus on developing methodologies that enable the recovery ofphoto-$z$ likelihoods with support over all redshifts, either directly or via aknown prior probability density.

How to obtain the redshift distribution from probabilistic redshift estimates