The Type Ia Supernova Color-Magnitude Relation and Host Galaxy Dust: A
  Simple Hierarchical Bayesian Model
  Conventional Type Ia supernova (SN Ia) cosmology analyses currently use a
simplistic linear regression of magnitude versus color and light curve shape,
which does not model intrinsic SN Ia variations and host galaxy dust as
physically distinct effects, resulting in low color-magnitude slopes. We
construct a probabilistic generative model for the dusty distribution of
extinguished absolute magnitudes and apparent colors as the convolution of a
intrinsic SN Ia color-magnitude distribution and a host galaxy dust
reddening-extinction distribution. If the intrinsic color-magnitude ($M_B$ vs.
$B-V$) slope $\beta_{int}$ differs from the host galaxy dust law $R_B$, this
convolution results in a specific curve of mean extinguished absolute magnitude
vs. apparent color. The derivative of this curve smoothly transitions from
$\beta_{int}$ in the blue tail to $R_B$ in the red tail of the apparent color
distribution. The conventional linear fit approximates this effective curve
near the average apparent color, resulting in an apparent slope $\beta_{app}$
between $\beta_{int}$ and $R_B$. We incorporate these effects into a
hierarchical Bayesian statistical model for SN Ia light curve measurements, and
analyze a dataset of SALT2 optical light curve fits of 248 nearby SN Ia at z <
0.10. The conventional linear fit obtains $\beta_{app} \approx 3$. Our model
finds a $\beta_{int} = 2.3 \pm 0.3$ and a distinct dust law of $R_B = 3.8 \pm
0.3$, consistent with the average for Milky Way dust, while correcting a
systematic distance bias of $\sim 0.10$ mag in the tails of the apparent color
distribution. Finally, we extend our model to examine the SN Ia luminosity-host
mass dependence in terms of intrinsic and dust components.
