AP Statistics Curriculum 2007 Bayesian Hierarchical

Sometimes we cannot be sure about the factuality of our prior knowledge. Often we make one or more assumptions about the relationships between the different unknown parameters \(\theta\) from which observations x has density p(x|\(\theta\)). These associations are sometimes referred to as structural. In some cases the structural prior knowledge is combined with a standard form of Bayesian prior belief about the parameters of the structure. In the case where \(\theta_i\) are independently and identically distributed, their common distribution might depend on a parameter \(\eta\) which we refer to as a hyperparameter; when the \(\eta\) is unknown we have a second tier in which we suppose we have a hyperprior p(\(\eta\)) expressing our beliefs about possible values of \(\eta\). In such a case we may say that we have a hierarchical model.