  A mild correlation exists in active galaxies between the mean black hole
accretion, as traced by the mean X-ray luminosity $\left<{\rm L}_{\rm
X}\right>$, and the host galaxy stellar mass M$_*$, characterised by a
normalisation steadily decreasing with cosmic time and lower in more quiescent
galaxies. We create comprehensive semi-empirical mock catalogues of active
black holes to pin down which parameters control the shape and evolution of the
$\left<{\rm L}_{\rm X}\right>-{\rm M}_*$ relation of X-ray detected active
galaxies. We find that the normalisation of the $\left<{\rm L}_{\rm
X}\right>-{\rm M}_*$ relation is largely independent of the fraction of active
galaxies (the duty cycle), but strongly dependent on the mean Eddington ratio,
when adopting a constant underlying M$_{\rm BH}-{\rm M}_*$ relation as
suggested by observational studies. The data point to a decreasing mean
Eddington ratio with cosmic time and with galaxy stellar mass at fixed
redshift. Our data can be reproduced by black holes and galaxies evolving on
similar M$_{\rm BH}-{\rm M}_*$ relations but progressively decreasing their
average Eddington ratios, mean X-ray luminosities, and specific star formation
rates, when moving from the starburst to the quiescent phase. Models consistent
with the observed $\left<{\rm L}_{\rm X}\right>-{\rm M}_*$ relation and
independent measurements of the mean Eddington ratios, are characterised by
M$_{\rm BH}-{\rm M}_*$ relations lower than those derived from dynamically
measured local black holes. Our results point to the $\left<{\rm L}_{\rm
X}\right>-{\rm M}_*$ relation as a powerful diagnostic to: 1) probe black
hole-galaxy scaling relations and the level of accretion onto black holes; 2)
efficiently break the degeneracies between duty cycles and accretion rates in
cosmological models of black holes.
