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Conventional Type Ia supernova (SN Ia) cosmology analyses currently use asimplistic linear regression of magnitude versus color and light curve shape,which does not model intrinsic SN Ia variations and host galaxy dust asphysically distinct effects, resulting in low color-magnitude slopes. Weconstruct a probabilistic generative model for the dusty distribution ofextinguished absolute magnitudes and apparent colors as the convolution of aintrinsic SN Ia color-magnitude distribution and a host galaxy dustreddening-extinction distribution. If the intrinsic color-magnitude ($M_B$ vs.$B-V$) slope $\beta_{int}$ differs from the host galaxy dust law $R_B$, thisconvolution results in a specific curve of mean extinguished absolute magnitudevs. 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 colordistribution. The conventional linear fit approximates this effective curvenear the average apparent color, resulting in an apparent slope $\beta_{app}$between $\beta_{int}$ and $R_B$. We incorporate these effects into ahierarchical Bayesian statistical model for SN Ia light curve measurements, andanalyze 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 modelfinds a $\beta_{int} = 2.3 \pm 0.3$ and a distinct dust law of $R_B = 3.8 \pm0.3$, consistent with the average for Milky Way dust, while correcting asystematic distance bias of $\sim 0.10$ mag in the tails of the apparent colordistribution. Finally, we extend our model to examine the SN Ia luminosity-hostmass dependence in terms of intrinsic and dust components.

The Type Ia Supernova Color-Magnitude Relation and Host Galaxy Dust: A Simple Hierarchical Bayesian Model