  We fit theoretical models to the emission-line spectra of 2766 Seyfert
galaxies from the Sloan Digital Sky Survey (SDSS). The model line fluxes are
derived by `mixing' photoionization model predictions of active galactic
nucleus (AGN) narrow line region (NLR) emission and HII region emission. The
observed line fluxes in each spectrum were directly compared to the grid of
mixed models using the Bayesian parameter estimation code NebulaBayes, thereby
measuring the degree of mixing in each spectrum for the first time. We find
that the majority of the Balmer line emission in the majority of
Seyfert-classified SDSS spectra arises from contaminating HII regions within
the fixed-size aperture. Even for spectra with log [OIII]/Hb $\gtrsim 0.9$, a
median of ~30% of the Balmer flux arises in HII regions. We show that the
results are qualitatively insensitive to the assumed Seyfert ionizing
continuum, and that ionizing spectra with a peak energy of E_peak ~ 40-50 eV
produce the most plausible distributions of mixing fractions. The analysis
cleanly quantifies how the starburst - AGN `mixing fraction' increases on the
BPT diagram for SDSS galaxies. Apart from the mixing fraction, the models also
vary with the ionization parameter in the NLR, the gas pressure, and the
metallicity. Measurements for the metallicity in particular will be presented
in a companion paper.
