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

Let $\theta =(\beta ,\pi )$ and regard $X_U$ as the latent variable. At iteration $t$, the E-step formsThe M-step updatesThis is the expectation-maximization algorithm for the posterior objective: including $\log p(\theta )$ in the complete-data log density makes the maximizer a maximum a posteriori estimate rather than a maximum-likelihood estimate.