  Improving the use of Type Ia supernovae (SNIa) as standard candles requires a
better approach to incorporate the relationship between SNIa and the properties
of their host galaxies. Using a spectroscopically-confirmed sample of
$\sim$1600 SNIa, we develop the first empirical model of underlying populations
for SNIa light-curve properties that includes their dependence on host-galaxy
stellar mass. These populations are important inputs to simulations that are
used to model selection effects and correct distance biases within the BEAMS
with Bias Correction (BBC) framework. Here we improve BBC to also account for
SNIa-host correlations, and we validate this technique on simulated data
samples. We recover the input relationship between SNIa luminosity and
host-galaxy stellar mass (the mass step, $\gamma$) to within 0.004 mags, which
is a factor of 5 improvement over the previous method that results in a
$\gamma$-bias of ${\sim}0.02$. We adapt BBC for a novel dust-based model of
intrinsic brightness variations, which results in a greatly reduced mass step
for data ($\gamma = 0.017 \pm 0.008$), and for simulations ($\gamma =0.006 \pm
0.007$). Analysing simulated SNIa, the biases on the dark energy
equation-of-state, $w$, vary from $\Delta w = 0.006(5)$ to $0.010(5)$ with our
new BBC method; these biases are significantly smaller than the $0.02(5)$
$w$-bias using previous BBC methods that ignore SNIa-host correlations.

Cosmology and Nongalactic Astrophysics