Posterior Distribution

Of interest to the modeller, then, is the conditional distribution of the parameters, given the data, that is . Bayes Theorem for random variables [27] yields

The distribution is termed the posterior distribution and describes the current state of knowledge about , given the initial knowledge of , together with the model, such knowledge having been updated by information. The constant of proportionality in the above is just where p(X) can be obtained from .

The Bayesian method, is then, quite straightforward [16]:

• construct a model, obtaining a likelihood ;
• elicit a prior distribution ;
• derive the posterior density as above.

In practice these tasks can be difficult to implement, and more is said about the details later.