A probabilistic formulation
Any regression problem can be expressed as an implementation of a probabilistic formulation. For instance what we typically have at our hand is a dependent variable y, a matrix X of covariates and a parameter vector β. The dependent variable consists of data we would like to learn something about or be able to explain. As such we wish to model it’s dynamics via the β through X. The joint probability distribution for these three ingredients is given simply as p(y, X,β). This is the most general form of representing a regression problem probabilistically. However, it’s not very useful, so in order to make it a bit more tangible let’s decompose this joint probability like this.