  The total extragalactic $\gamma$-ray flux provides a powerful probe into the
origin and evolution of the highest energy processes in our universe. An
important component of this emission is the isotropic $\gamma$-ray background
(IGRB), composed of sources that cannot be individually resolved by current
experiments. Previous studies have determined that the IGRB can be dominated by
either misaligned active galactic nuclei (mAGN) or star-forming galaxies
(SFGs). However, these analyses are limited, because they have utilized binary
source classifications and examined only one source class at a time. We perform
the first combined joint-likelihood analysis that simultaneously correlates the
$\gamma$-ray luminosity of extragalactic objects with both star-formation and
mAGN activity. We find that SFGs produce 48$^{+33}_{-20}$\% of the total IGRB
at~1 GeV and 56$^{+40}_{-23}$\% of the total IGRB at 10 GeV. The contribution
of mAGN is more uncertain, but can also be significant. Future work to quantify
the radio and infrared properties of nearby galaxies could significantly
improve these constraints.
