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The scientific impact of current and upcoming photometric galaxy surveys iscontingent on our ability to obtain redshift estimates for large numbers offaint galaxies. In the absence of spectroscopically confirmed redshifts,broad-band photometric redshift point estimates (photo-$z$s) have beensuperseded by photo-$z$ probability density functions (PDFs) that encapsulatetheir nontrivial uncertainties. Initial applications of photo-$z$ PDFs in weakgravitational lensing studies of cosmology have obtained the redshiftdistribution function $\mathcal{N}(z)$ by employing computationallystraightforward stacking methodologies that violate the laws of probability. Inresponse, mathematically self-consistent models of varying complexity have beenproposed in an effort to answer the question, "What is the right way to obtainthe redshift distribution function $\mathcal{N}(z)$ from a catalog of photo-$z$PDFs?" This letter aims to motivate adoption of such principled methods byaddressing the contrapositive of the more common presentation of such models,answering the question, "Under what conditions do traditional stacking methodssuccessfully recover the true redshift distribution function $\mathcal{N}(z)$?"By placing stacking in a rigorous mathematical environment, we identify twosuch conditions: those of perfectly informative data and perfectly informativeprior information. Stacking has maintained its foothold in the astronomicalcommunity for so long because the conditions in question were only weaklyviolated in the past. These conditions, however, will be strongly violated byfuture galaxy surveys. We therefore conclude that stacking must be abandoned infavor of mathematically supported methods in order to advance observationalcosmology.

physical review. d/physical review. d.

How not to obtain the redshift distribution from probabilistic redshift  estimates: Under what conditions is it not inappropriate to estimate the  redshift distribution N(z) by stacking photo-z PDFs?