Let us assume that the noise terms and in Equation (5.18) are all independent at different time instants and Gaussian with zero mean. The different components of the vector are, however, allowed to have different variances as governed by their hyperparameters. Following the developments of Section 3.1.1, the model implies a likelihood for the data

where denotes a diagonal matrix with the elements of the vector on the diagonal. The vector is a hyperparameter that defines the variances of different components of the noise.

As the values of the noise at different time instants are independent, the full data likelihood can be written as

where the individual factors are of the form