$\beta$ and $m_0$, we calculate the UV apparent magnitude limit $m_{\mathrm{UV}}$ as that at rest-frame 1500Å, or $\lambda=(1+z)\times0.15\mu$m. We convert this to an absolute magnitude using $M_{\mathrm{UV}}=m_{\mathrm{UV}}-DM+2.5\log(1+z)$ where $DM$ is the distance modulus.

We adopt the $z=6$ $M_\ast-M_{\mathrm{UV}}$ relation derived by @Song2016 using a large sample of galaxies from the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS)/Great Observatories Origins Deep Survey (GOODS) fields and the Hubble Ultra Deep Field (HUDF): $$\label{SongRelation}
\log(M_\ast) = 9.53 \pm 0.02 - (0.50 \pm 0.03)\times(M_{\mathrm{UV}}+21)$$ with a scatter of 0.36 dex. We use a Monte Carlo technique, sampling from a uniform distribution of magnitudes ranging from $M_{\mathrm{UV}}=-20$ mag to our $2\sigma$ upper limit for each quasar host. The lower luminosity limit of $M_{\mathrm{UV}}=-20$ mag was chosen as this is as faint as high-redshift quasar hosts are expected to be from the BlueTides simulation [@Feng2015; @Marshall2019 see Figure \[fig:BT\_comparison\]]. We assign a stellar mass to each sampled magnitude using Equation \[SongRelation\], given a normal distribution with $\sigma=0.36$ dex, to determine the resulting probability distribution of stellar masses. These stellar masses are normally-distributed, so we adopt the 2$\sigma$ upper limit from this relation as our host mass limit. We note that this results in a lower, 