Data Analysis Recipes: Products of multivariate Gaussians in Bayesian inferences

Understand

A product of two Gaussians (or normal distributions) is another Gaussian.

• That's a valuable and useful fact! Here we use it to derive a refactoring of a common product of multivariate Gaussians: The product of a Gaussian likelihood times a Gaussian prior, where some or all of those parameters enter the likelihood only in the mean and only linearly.
• That is, a linear, Gaussian, Bayesian model.
• This product of a likelihood times a prior pdf can be refactored into a product of a marginalized likelihood (or a Bayesian evidence) times a posterior pdf, where (in this case) both of these are also Gaussian.