2.3.3 Bayes' Theorem for Gaussian Variables - Pattern Recognition and Machine Learning

We consider the situation where we're given a Gaussian prior and a Gaussian likelihood where our observations are affinely related to the variables of interest. We complete the square to determine the joint distribution of the variables of interest and the observations and use our earlier work on jointly Gaussian variables to determine the marginal distributions of the observations and the posterior distribution of the variables of interest. I conclude by providing some intuition on where the expression for the posterior distribution comes from by framing it as combining two Gaussian sources of information about the variables of interest.