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

Ignoring constants, the complete-data log posterior isTake conditional expectations under $({\beta }^{(t)},{\pi }^{(t)})$. DefineCompleting the square givesFor $v\ne 0$,so $A$ is positive definite. Exact rowwise expectations require summing over $2^{|U_i|}$ states. Thus the cost is exponential in the largest number of missing covariates in one row, more precisely ${\sum }_i{2}^{|U_i|}$ times a polynomial factor for accumulating first and second moments.