and define the function $Q(\theta;\theta_{k})=\mathbb{E}_{U|X=x,\theta_{k}}[L(\theta;x,U)]$.

-   **Maximization Step (M-step)**: Update the estimate of $\theta$ by maximizing the $Q$ function: $\theta_{k+1}=\arg\max_{\theta}\,Q(\theta;\theta_{k})$.

The EM algorithm has a remarkable monotonicity property, *i.e.*, the marginal log-likelihood $\ell(\theta;x)$ is always nondecreasing on the $\{\theta_{k}\}$ sequence. Due to such nice properties, the EM algorithm has been the standard optimization technique for Gaussian mixture models and many other missing data models. However, one serious problem of the EM algorithm is that the expectation defining the $Q$ function usually has no simple closed form, so the Monte Carlo EM algorithm (MCEM, [@wei1990monte]; [@levine2001implementations]) proposes to use Monte Carlo methods to approximate the expectation. Using MCMC to approximate the expectation in the E-step is not a new idea, but what really matters is how to properly choose the hyperparameters to guarantee the convergence of the M-step.

In this sense, Theorem \[thm:convergence\_convex\] provides a clear way to make the EM algorithm effectively automated. It is easy to see that the target distribution $\pi(\xi)$ is $p(u|x;\theta_{k})$, the conditional density of $U$ given $X=x$, which is proportional to the joint density of $(X,U)$ under $\theta_{k}$. Therefore, we can define $V(\xi)=-L(\theta_{k};x,\xi)$, and then apply Algorithm \[alg:sagd\] to directly solve the M-step, whose convergence 