Normal distribution belongs in exponential family, so in its log PDF the 2 terms involving \(x\) has fixed form, which can be useful for deriving various identities.
For multivariate normal distribution:
Let weight distribution be \(N(\theta|\mu,\Sigma)\). Then
\[\begin{align} A\theta\sim N(A\mu,A\Sigma A^{T}) \end{align} \]Let prior be \(N(\theta|\mu,\Sigma)\), likelihood be \(N(y|x^{T}\theta,\Sigma_{s})\). Then the posterior of \(\theta\) can be written as
\[\begin{equation} \theta|x \sim N((\Sigma^{-1}+x\Sigma_{s}^{-1}y)^{-1}(\Sigma^{-1}\mu+x\Sigma_{s}^{-1}y), \Sigma^{-1}+x\Sigma_{s}^{-1}y) \end{equation} \]Let weight distribution be \(N(\theta|\mu,\Sigma)\), likelihood be \(N(y|x^{T}\theta,\Sigma_{s})\). Then the predictive distribution of \(x\) can be written as
\[\begin{equation} x|\theta \sim N(x^{T}\mu,x^{T}\Sigma x+\Sigma_{s}) \end{equation} \]Let distribution be \(N(x|\mu,\Sigma)\). Split \(x\) into two half \(x_{1}\) and \(x_{2}\), and split \(\mu\) and \(\Sigma\) accordingly. (The first row of \(\Sigma\) contains \(\Sigma_{11}\) and \(\Sigma_{12}\).) The conditional distribution of \(x_{2}\) given \(x_{1}\) with values \(v_{1}\) can be written as
\[\begin{align} x_{2}|(x_{1}=v_{1}) &\sim N(\mu_{2|1},\Sigma_{2|1})\\ \mu_{2|1} &= \mu_{2} + \Sigma_{21}\Sigma_{22}^{-1}(v_{1}-\mu_{1})\\ \Sigma_{2|1} &= \Sigma_{2} - \Sigma_{21}\Sigma_{11}^{-1}\Sigma_{12} \end{align} \]