21
Bayesian regression analysis
In Bayesian inference the parameters θ are also considered as stochastic quantities 
with corresponding probability distribution π(·), called a priori distribution. In the case of continuous parameters the a priori distribution is determined by a probability density π(·) on the parameter space
Then π(·) is a probability density on the parameter space, called a priori density.
The stochastic model for the dependent variable y is Yx ~ fx(·|θ).
21.1 Calculation of a posteriori distributions
For observed data (xi, yi), i = 1(1)n the likelihood function 
(θ; (x1, y1),…,(xn, yn)) is given by
in the case of independent observations y1,…,yn.
The a posteriori density π(·|(x1, y1),…,(xn, yn)) of is given by Bayes’ theorem which reads here
In extended form, Bayes’ theorem takes the following form:
Here the parameter θ = (θ1,…,θk) ∈ k can be k-dimensional. Then the ...
