Gaussian conjugacy for a normal linear model

If $Y\mid \beta \sim N(X\beta ,{\Sigma }_e)$ and $\beta \sim N(m_0,{\Sigma }_0)$, then the posterior is normal with precision matrix $X^T{\Sigma }_e^{-1}X+{\Sigma }_0^{-1}$ and mean equal to the inverse precision times $X^T{\Sigma }_e^{-1}Y+{\Sigma }_0^{-1}{m}_0$.

A Gaussian likelihood is a likelihood function obtained by modeling the observations conditionally with a normal distribution.