Cartoon line-of-sight reddening profile assuming a model with a single velocity slice v₁ (where the cloud can be described by a single distance component d₁). Such a reddening profile is defined by the free parameters d_fore (foreground cloud distance), E_fore (foreground cloud reddening), d₁ (distance modulus to single velocity slice), and c₁ (gas-to-dust conversion coefficient in single velocity slice). The quantity I_j,1 constitutes the CO emission for star i lying in pixel j in the single velocity slice v₁. The reddening profile is overlaid on an idealized joint posterior on distance and reddening (grayscale ellipsoid) for individual star i lying in CO pixel j. The star’s likelihood is simply the integral over μ following the reddening profile (solid red line); the dashed red lines are excluded from the integration. In Section 6 the likelihood from this star would be multiplied together with the likelihoods of thousands of other stars to get the total likelihood contribution (see Equation (4)).