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We present a rigorous mathematical solution to photometric redshift estimation and the more general inversion problem. The challenge we address is to meaningfully constrain unknown properties of astronomical sources based on given observables, usually multicolor photometry, with the help of a training set that provides an empirical relation between the measurements and the desired quantities. We establish a formalism that blurs the boundary between the traditional empirical and template-fittingdoi:10.1088/0004-637x/695/1/747 fatcat:jj7ekn5vrfbu3n7ipotlbs3lmq