# AP Statistics Curriculum 2007 Bayesian Hierarchical

(Difference between revisions)
 Revision as of 05:58, 2 June 2009 (view source)JayZzz (Talk | contribs) (New page: 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 $\theta...)← Older edit Revision as of 23:28, 22 October 2009 (view source)IvoDinov (Talk | contribs) Newer edit → Line 1: Line 1: + ==[[EBook | Probability and Statistics Ebook]] - Bayesian Hierarchical Models== + Sometimes we cannot be sure about the factuality of our prior knowledge. Often we make one or more assumptions about the relationships between the different unknown parameters [itex]\theta$ 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. 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. Line 11: Line 13: ==Bayesian analysis for unknown overall mean== ==Bayesian analysis for unknown overall mean== + + + ==See also== + * [[EBook#Chapter_III:_Probability |Probability Chapter]] + + ==References== + +

## 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 θ from which observations x has density p(x|θ). 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 θi are independently and identically distributed, their common distribution might depend on a parameter η which we refer to as a hyperparameter; when the η is unknown we have a second tier in which we suppose we have a hyperprior p(η) expressing our beliefs about possible values of η. In such a case we may say that we have a hierarchical model.