  The X-ray light-curves of the recurring outbursts observed in low-mass X-ray
binaries provide strong test beds for constraining (still) poorly understood
disc-accretion processes. These light-curves act as a powerful diagnostic to
probe the physics behind the mechanisms driving mass inflow and outflow in
these binary systems. We have thus developed an innovative methodology,
combining a foundation of Bayesian statistics, observed X-ray light-curves, and
accretion disc theory. With this methodology, we characterize the
angular-momentum (and mass) transport processes in an accretion disc, as well
as the properties of the X-ray irradiation-heating that regulates the decay
from outburst maximum in low-mass X-ray transients. We recently applied our
methodology to the Galactic black-hole low-mass X-ray binary population,
deriving from their lightcurves the first-ever quantitative measurements of the
$\alpha$-viscosity parameter in these systems \citep{tetarenko2018}. In this
paper, we continue the study of these binaries, using Bayesian methods to
investigate the X-ray irradiation of their discs during outbursts of strong
accretion. We find that the predictions of the disc-instability model, assuming
a source of X-ray irradiation proportional to the central accretion rate
throughout outburst, do not adequately describe the later stages of BH-LMXB
outburst light-curves. We postulate that the complex and varied light-curve
morphology observed across the population is evidence for irradiation that
varies in time and space within the disc, throughout individual transient
outbursts. Lastly, we demonstrate the robustness of our methodology, by
accurately reproducing the synthetic model light-curves computed from numerical
codes built to simulate accretion flows in binary systems.
