# AP Statistics Curriculum 2007 Bayesian Hierarchical

## Probability and Statistics Ebook - Bayesian Hierarchical Models

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 to 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.