- $grizy$ photometry
- Covers entire sky north of $\delta = -30^{\circ}$

- $J H K_s$ photometry
- Full-sky

~800 million stars with high-quality, multi-band photometry

Assume all stars in one pixel trace the same reddening column

$\hphantom{\mu_{k} ,\,} \Delta E_{j} ,$ | $\ \ j = 1, \ldots , n_{\mathrm{bins}}$ |

$\mu_{k} ,\,$ $\delta_{k} ,\,$ $\Theta_{k} ,$ | $\ \ k = 1, \ldots , n_{\mathrm{stars}}$ |

$$
\left.
\vphantom{
\begin{align}
\Delta E_{j} \\
\Theta_{k} \\
\Theta_{k}
\end{align}
}
\right\} \Rightarrow \vec{m}_{k}
$$

$p \left( \left\{ \Delta E \right\} | \left\{ \vec{m} \right\} \right)$
$\propto p \left( \left\{ \Delta E \right\} \right)$
$\prod_{k}$
$\int$
$\mathrm{d} \mu_{k}$
$\mathrm{d} \Theta_{k}$
$\mathrm{d} \delta_{k}$
$p \left( \vec{m}_{k} \, | \, \mu_{k} , \Theta_{k} , \left\{ \Delta E \right\} , \delta_{k} \right)$

$\times \ p \left( \mu_{k} , \Theta_{k}, \delta_{k} \, | \, \left\{ \Delta E \right\} \right)$

$\tilde{p} \left( \mu_{k} , E \, | \, \vec{m}_{k} \right) = $
$\int \mathrm{d} \delta_{k}$
$\int \mathrm{d} \Theta_{k}$
$p \left( \mu_{k} , \Theta_{k} , \vphantom{\left( 1 + \delta_{k} \right)} \right.$
$\left( 1 + \delta_{k} \right)$
$\left. E \, | \, \vec{m}_{k} \right)$
$p \left( \delta_{k} \, | \, E \right)$

$p \left( \left\{ \Delta E \right\} | \left\{ \vec{m} \right\} \right)$
$\propto p \left( \left\{ \Delta E \right\} \right)$
$\prod_{k}$
$\int \mathrm{d} \mu_{k}$
$\tilde{p} \left( \mu_{k} , E \left( \mu_{k} , \left\{ \Delta E \right\} \right) | \, \vec{m}_{k} \right)$

$\hphantom{p \left( \left\{ \Delta E \right\} | \left\{ \vec{m} \right\} \right) \propto p \left( \left\{ \Delta E \right\} \right) \prod_{k} \int \mathrm{d} \mu_{k} \ } \underbrace{\hphantom{ \tilde{p} \left( \mu_{k} , E \left( \mu_{k} , \left\{ \Delta E \right\} \right) | \, \vec{m}_{k} \right) }}_{\mathrm{pre-computed}}$

Mathematically equivalent to parameterization shown earlier.

Each star puts a constraint on the line-of-sight reddening.

Hundreds of stars per sightline.

The cross-sightline differences in the above map are not *a priori* likely.

- Should include correlations between dust in neighboring sightlines, assuming some dust density power spectrum.
- Couples stellar reddenings across the sky.
- Since information is shared naturally across sightlines, we could theoretically shrink sightlines arbitrarily without loss of information.
- Computational difficulty of large, coupled problem.