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-fitting algorithms, as both are just special cases that are discussed in detail to put them in context. The new approach enables the development of more sophisticated methods that go beyond the classic techniques to combine their advantages. We look at the directions for further improvement in the methodology, and examine the technical aspects of practical implementations. We show how training sets are to be constructed and used consistently for reliable estimation.
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