and Section \[sec:accdist\] above).

Model A
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We begin with a simple model (A), where we assume a single scaling applies between SMBH mass and the total stellar mass of the host galaxy at all redshifts. We take the scaling between SMBH mass and galaxy mass as ${\mathcal{M}_\mathrm{bh}}\approx 0.002{\mathcal{M}_*}$ from @Marconi03, where we have further assumed that ${\mathcal{M}_*}\approx \mathcal{M}_\mathrm{bulge}$. Thus, we can directly translate the specific accretion rate into an Eddington ratio for the SMBH (see Equation \[eq:ledd\]). With this scaling assumed, we apply a strict cut to our specific accretion rate distribution at the Eddington limit, $\log \lambda=0$, as shown in Figure \[fig:accdists\] (left panel). The grey region tracks the uncertainty in our specific accretion rate distribution. We estimate the uncertainties using the errors in the best-fit model parameters from @Aird12. However, these errors assume a fixed model form and do not fully account for the observational uncertainties in the measurements of the $\lambda$ distribution at a given stellar mass or redshift. We therefore increase the uncertainty in the overall normalization by $\sim \pm 0.2$ dex, which provides better agreement between our model uncertainty and the errors in the individual binned measurements of $p(\lambda {\;|\;}{\mathcal{M}_*},z)$ in figure 7 of 