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 above integral is
The a posteriori density carries all information concerning the parameter vector θ = (θ1,…,θk) which comes from the a priori distribution and the data.