To use the techniques of probabilistic modelling, one must usually somehow specify the likelihood of the data given some parameters. Not all models are, however, probabilistic in nature and have naturally emerging likelihoods. Many models are defined as generative models by stating how the observed data could be generated from unknown parameters or latent variables.
For given data vectors , a simple linear generative model could be written as
One way to turn such a generative model into a probabilistic model is to add a noise term to the Equation (3.4). This yields
This implies a likelihood for the data given by
For a complete probabilistic model one also needs priors for all the parameters of the model. For instance hierarchical models and conjugate priors can be useful mathematical tools here but in the end the priors should be chosen to represent true prior knowledge on the solution of the problem at hand.