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:

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