8.1 The idea of a hierarchical model

8.1.1 Definition

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

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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 ...

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