Solution
= Solution
The stated conditional independences give
$$
p(\mathbf d\mid\mathbf x_i,m_i)=
\prod_{s=1}^{N_{\mathrm{sat}}}
\frac1{\sqrt{2\pi\sigma_s^2}}
\exp\left[-\frac{(d_s-x_{si})^2}{2\sigma_s^2}\right].
$$
Thus
$$
\widehat{\mathbb E[m\mid\mathbf d]}=\sum_{i=1}^Km_iw_i,
\qquad
w_i=\frac{\prod_s p(d_s\mid x_{si})}
{\sum_j\prod_s p(d_s\mid x_{sj})}.
$$
The dependence among satellite properties and host mass remains encoded in each jointly simulated row.