Solution
= Solution
All three sets of statistics are independent. With $s_1=\widehat S_1$, $s_{2g}=\widehat S_2^g$, and $s_{3i}=\widehat S_3^i$, their <likelihood function>, up to a parameter-independent factor, is
$$
L(M_0^*,\Delta M,\mathcal M)
\propto
\exp\!\left[-\frac{(s_1-M_0^*)^2}{2V_1}
-\sum_{g=1}^G\frac{(s_{2g}-\Delta M)^2}{2V_2}
-\sum_{i=1}^{N_{\mathrm{SN}}}\frac{(s_{3i}-\mathcal M)^2}{2\sigma_{\mathrm{SN}}^2}
\right].
$$