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?

The scientific impact of current and upcoming photometric galaxy surveys is contingent on our ability to obtain redshift estimates for large numbers of faint galaxies. In the absence of spectroscopically confirmed redshifts, broad-band photometric redshift point estimates (photo-$z$s) have been superseded by photo-$z$ probability density functions (PDFs) that encapsulate their nontrivial uncertainties. Initial applications of photo-$z$ PDFs in weak gravitational lensing studies of cosmology have obtained the redshift distribution function $\mathcal{N}(z)$ by employing computationally straightforward stacking methodologies that violate the laws of probability. In response, mathematically self-consistent models of varying complexity have been proposed in an effort to answer the question, "What is the right way to obtain the redshift distribution function $\mathcal{N}(z)$ from a catalog of photo-$z$ PDFs?" This letter aims to motivate adoption of such principled methods by addressing the contrapositive of the more common presentation of such models, answering the question, "Under what conditions do traditional stacking methods successfully recover the true redshift distribution function $\mathcal{N}(z)$?" By placing stacking in a rigorous mathematical environment, we identify two such conditions: those of perfectly informative data and perfectly informative prior information. Stacking has maintained its foothold in the astronomical community for so long because the conditions in question were only weakly violated in the past. These conditions, however, will be strongly violated by future galaxy surveys. We therefore conclude that stacking must be abandoned in favor of mathematically supported methods in order to advance observational cosmology.