Directed acyclic graph describing the hierarchical Bayesian model for colors and spectroscopic velocities. The open boxes represent the unknown parameters of individual supenovae (the dust extinction and intrinsic colors ) and the unknown hyperparameters of the SN Ia population () and the dust population (τ). The shaded boxes represent the measured data: the spectroscopic velocity measurement v_s and the observed colors . The arrows represent links of conditional probability relating the hyperparameters, parameters, and the data (green: intrinsic effects; red: dust effects; purple: measurement error).