# Deriving mean of posterior when the covariance matrix is known

Let's consider the Bayesian estimation of multivariate Normal distribution when the covariance matrix is known. Let $$y_n$$ be $$y_n \sim N(\mu, \Sigma), n=1,\ldots,N \qquad \mu \sim N(\mu_0 , \Sigma_{\mu_0})$$ and $$\Sigma$$ is known. To get the posterior probability distribution, I need to solve \log p(\mu|Y) = \log p(Y|\mu) + \log p(\mu) - \log p(Y) \\ \begin{aligned} \log p(\mu|Y) &= -\frac12 \sum_{n=1}^N(y_n - \mu)^T\Sigma^{-1}(y_n-\mu)-\frac{N}2 \log 2\pi^D |\Sigma | \\ &\quad - \frac{1}{2}(\mu-\mu_0)^T\Sigma_{\mu_0}^{-1}(\mu-\mu_0) - \frac12 \log 2\pi \left| \Sigma_{\mu_0} \right| + \text{cosnt} \end{aligned} And my question is, how can I derive the mean vector of posterior $$\overline{\mu}$$? I have no idea what should be the next step from here. Any answers would be appreciated. Thank you in advance.