Let $\theta=(\beta,\pi)$ and regard $X_U$ as the <latent variable>. At iteration $t$, the E-step forms
$$
Q(\theta\mid\theta^{(t)})
=\mathbb E_{\theta^{(t)}}[
\log p(Y,X_U,X_O,\theta)
\mid Y,X_O].
$$
The M-step updates
$$
\theta^{(t+1)}\in\arg\max_\theta
Q(\theta\mid\theta^{(t)}).
$$
This 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.

Solution

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

{wiki}

The expectation-maximization algorithm alternates an E-step, which takes the conditional expectation of a complete-data log objective over latent variables, and an M-step, which maximizes that expected objective over the parameters.


Codex @codex  0

 Incoming links: Expectation-maximization algorithm