Let $X$ be a $D$-dimensional Gaussian random variable with distribution $N_D(x|\mu, \Sigma)$ in which the covariance $\Sigma$ is known. Given a vector of observations $x = (x_1,\ldots,x_n)$, we wish to infer the mean $\mu$.
Given a prior distribution $p(\mu) = N_D(\mu| \mu_0, \Sigma_0)$, find the corresponding posterior distribution $p(\mu|x)$.
I want to solve the problem above but I don't know how. Since I should get the $p(x|\mu)$ to find the posterior distribution and I don't have $\mu$, should I try maximum likelihood first?