  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.
