8.1 The idea of a hierarchical model
So far, we have assumed that we have a single known form to our prior distribution. Sometimes, however, we feel uncertain about the extent of our prior knowledge. In a typical case, we have a first stage in which observations have a density which depends on r unknown parameters
for which we have a prior density . Quite often we make one or more assumptions about the relationships between the different parameters , for example, that they are independently and identically distributed [sometimes abbreviated i.i.d.] or that they are in increasing order. Such relationships are often 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. Thus, in the case where the are independently and identically distributed, their common distribution ...