= Solution
Both <EM algorithm> and <data augmentation> exploit a latent completion $z$, but they use it differently. EM takes a <conditional expectation> of the complete-data <log-likelihood> and then maximizes over the parameter, producing a deterministic sequence of point estimates when the M-step is uniquely specified.
In Bayesian <data augmentation>, choose a <prior distribution> $p(\theta)$ and target
$$
p(\theta,z\mid x)\propto f(x,z;\theta)p(\theta).
$$
Alternate the <Gibbs sampler> updates
$$
z^{(t+1)}\sim p(z\mid x,\theta^{(t)}),\qquad\theta^{(t+1)}\sim p(\theta\mid x,z^{(t+1)}).
$$
These are stochastic draws from <full conditional distributions>, not an expectation followed by maximization. Their invariant law is the augmented joint <posterior>, and discarding $z$ gives <posterior> inference for $\theta$, including uncertainty rather than only an MLE. A sampled completion followed by maximization would instead be a stochastic EM variant. Individual <likelihood> values along a data-augmentation chain need not increase: the chain is designed to sample a <posterior>, not to ascend the <likelihood>.
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