Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 216 / 4 / b / Solution

Let $U_i=\{j:(i,j)\notin O\}$ and let $x_{i,O}$ denote the observed entries in row $i$. Conditional on the parameters, different rows of the missing design are independent, while the missing entries within one row are coupled by its Gaussian response. For $z\in \{0,1{\}}^{U_i}$,Normalizing this expression over the $2^{|U_i|}$ configurations and multiplying over rows gives the full conditional distribution of $X_U$.