The WISE mission has unveiled a rare population of high-redshift ($z=1-4.6$), dusty, hyper-luminous galaxies, with infrared luminosities $L_{rm IR} > 10^{13}~L_{odot}$, and sometimes exceeding $10^{14}~L_{odot}$. Previous work has shown that their dust temperatures and overall far-IR SEDs are significantly hotter than expected for star-formation. We present here an analysis of the rest-frame optical through mid-IR SEDs for a large sample of these so-called Hot, Dust-Obscured Galaxies (Hot DOGs). We find that the SEDs of Hot DOGs are generally well modeled by the combination of a luminous, yet obscured AGN that dominates the rest-frame emission at $lambda > 1murm m$ and the bolometric luminosity output, and a less luminous host galaxy that is responsible for the bulk of the rest optical/UV emission. Even though the stellar mass of the host galaxies may be as large as $10^{11}-10^{12}~M_{odot}$, the AGN emission, with luminosities comparable to those of the most luminous QSOs known, require that either Hot DOGs have black hole masses significantly in excess of the local relations, or that they radiate significantly above the Eddington limit. We show that, while rare, the number density of Hot DOGs is comparable to that of equally luminous but unobscured (i.e., Type 1) QSOs. This is inconsistent with the trend of a diminishing fraction of obscured objects with increasing luminosity found for less luminous QSOs, possibly indicating a reversal in this relation at high luminosity, and that Hot DOGs are not the torus-obscured counterparts of the known optically selected, largely unobscured Hyper-Luminous QSOs. Hot DOGs may represent a different type of galaxy and thus a new component of the galaxy evolution paradigm. Finally, we discuss the environments of Hot DOGs and show that these objects are in regions as dense as those of known high-redshift proto-clusters.(Abridged)
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