I'm trying to understand applying the EM algorithm to compute the MLE in a missing data problem.
Specifically, suppose $(x_1,y_1),\ldots,(x_n,y_n)$ is a random sample from the bivariate normal distribution with mean $(0,0)$ and unknown covariance matrix $$ \Sigma=\begin{bmatrix} \sigma_x^2&\rho\sigma_x\sigma_y\\ \rho\sigma_x\sigma_y&\sigma_y^2 \end{bmatrix}. $$ I want to find the MLE of $\Sigma$ given that the first $\nu< n$ of the $y$-coordinates are missing.
For the E-step of the EM algorithm, I need to compute
$$ Q_i(\Sigma,\tilde\Sigma):= \begin{cases} \mathbb{E}_{y\sim f(y\mid x_i,\tilde\Sigma)}[\log f(x_i,y\mid\Sigma)]& \text{if }1\leq i\leq \nu,\\ \log f(x_i,y_i\mid\Sigma)&\text{if }\nu< i\leq n, \end{cases} $$ where $$ \begin{aligned} -\log f(x,y\mid\Sigma) = \log2\pi &+ \frac12\log\sigma_x^2 + \frac12\log\sigma_y^2 + \frac12\log(1-\rho^2)\\ &+ \frac{1}{2(1-\rho^2)} \left\{\frac{x^2}{\sigma_x^2} - \frac{2\rho xy}{\sigma_x\sigma_y} + \frac{y^2}{\sigma_y^2}\right\} \end{aligned} $$ and $$ \tilde\Sigma=\begin{bmatrix} \tilde\sigma_x^2&\tilde\rho\tilde\sigma_x\tilde\sigma_y\\ \tilde\rho\tilde\sigma_x\tilde\sigma_y&\tilde\sigma_y^2 \end{bmatrix}. $$
Since $$ \begin{aligned} \mathbb{E}_{y\sim f(y\mid x_i,\tilde\Sigma)} [y] &= \frac{\tilde\rho \tilde\sigma_yx_i}{\tilde\sigma_x},\\ \mathbb{E}_{y\sim f(y\mid x_i,\tilde\Sigma)}[y^2] &=\tilde\sigma_y^2(1-\tilde\rho^2) + \frac{\tilde\rho^2 \tilde\sigma^2_yx_i^2}{\tilde\sigma_x^2}, \end{aligned} $$
I get that
$$ \begin{aligned} -Q_i(\Sigma,\tilde\Sigma) = \log2\pi &+ \frac12\log\sigma_x^2 + \frac12\log\sigma_y^2 + \frac12\log(1-\rho^2)\\ &+ \frac{1}{2(1-\rho^2)} \left\{\frac{x_i^2}{\sigma_x^2} - \frac{2\rho\tilde\rho\tilde\sigma_y^2 x_i^2}{\sigma_x\tilde\sigma_x\sigma_y} + \frac{\tilde\sigma_x^2\tilde\sigma_y^2(1-\tilde\rho^2)+ \tilde\rho^2 \tilde\sigma^2_yx_i^2}{\tilde\sigma_x^2\sigma_y^2} \right\}. \end{aligned} $$
for $1\leq i\leq \nu$.
Now, for the M-step, I need to compute $$ \operatorname*{argmax}_{\Sigma} \sum_{i=1}^n Q_i(\Sigma,\tilde\Sigma). $$
And here I'm stuck. Is there a nice form for the optimal $\Sigma$?