The sources and production mechanisms of high-energy astrophysical neutrinos are largely unknown. A promising opportunity for progress lies in the study of neutrino flavor composition, i.e., the proportion of each flavor in the flux of neutrinos, which reflects the physical conditions at the sources. To seize it, we introduce a Bayesian method that infers the flavor composition at the neutrino sources based on the flavor composition measured at Earth. We find that present data from the IceCube neutrino telescope favor neutrino production via the decay of high-energy pions and rule out production via the decay of neutrons. In the future, improved measurements of flavor composition and mixing parameters may single out the production mechanism with high significance.