  We develop an effective field theory (EFT) for nuclear vibrations. The key
ingredients - quadrupole degrees of freedom, rotational invariance, and a
breakdown scale around the three-phonon level - are taken from data. The EFT is
developed for spectra and electromagnetic moments and transitions. We employ
tools from Bayesian statistics for the quantification of theoretical
uncertainties. The EFT consistently describes spectra and electromagnetic
transitions for $^{62}$Ni, $^{98,100}$Ru, $^{106,108}$Pd, $^{110,112,114}$Cd,
and $^{118,120,122}$Te within the theoretical uncertainties. This suggests that
these nuclei can be viewed as anharmonic vibrators.
