Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 219 / 2 / i / Solution

Write $v={\sigma }_{\mathrm{int}}^2$, $r_s={\sigma }_{m,s}^2$ and $d_s=\mu (z_s;H_0,w,{\Omega }_M)$. The latent absolute magnitude and measurement error are independent normal variables, so their convolution is normal. After integrating out the individual intrinsic magnitudes, each apparent magnitude has distributionConsequently the likelihood function for the independent sample isThis is the marginal observational likelihood of a hierarchical Bayesian model, not the likelihood conditional on each unknown intrinsic magnitude. The individual errors remain heteroscedastic through their known $r_s$.