21.6 Bayesian regression with Gaussian distributions
For linear regression models with vector independent variable x = (x1,…,xk),
and homoscedastic Gaussian distributions of the dependent variable Yx the likelihood function takes the following form:
If the variance σ2 is known we obtain
For unknown variance σ2 the likelihood is a function of σ2 also,
and the a priori density has to be a function of θ0,…,θk and σ2, i.e. π(θ0, θ1,…,θk, σ2).
The a posterior density is given by Bayes’ theorem: