A binning-free method reveals a continuous relationship between
  galaxies' AGN power and offset from main sequence
  Studies investigating the relationship between AGN power and the star
formation rates (SFRs) of their host galaxies often rely on averaging
techniques -- such as stacking -- to incorporate information from
non-detections. However, averages, and especially means, can be strongly
affected by outliers and can therefore give a misleading indication of the
"typical" case. Recently, a number of studies have taken a step further by
binning their sample in terms of AGN power (approximated by the 2-10keV
luminosity of the AGN), and investigating how the SFR distribution differs
between these bins. These bin thresholds are often weakly motivated, and
binning implicitly assumes that sources within the same bin have similar (or
even identical) properties. In this paper, we investigate whether the
distribution of host SFRs -- relative to the locus of the star-forming main
sequence (i.e., $R_{\rm MS}$) -- changes continuously as a function of AGN
power. We achieve this by using a hierarchical Bayesian model that completely
removes the need to bin in AGN power. In doing so, we find strong evidence that
the $R_{\rm MS}$ distribution changes with 2-10keV X-ray luminosity. The
results suggest that higher X_ray luminosity AGNs have a tighter physical
connection to the star-forming process than lower X-ray luminosity AGNs, at
least within the $0.8 < z < 1.2$ redshift range considered here.
