# AP Statistics Curriculum 2007 Bayesian Hierarchical

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+ | ==[[EBook | Probability and Statistics Ebook]] - Bayesian Hierarchical Models== | ||

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

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==Bayesian analysis for unknown overall mean== | ==Bayesian analysis for unknown overall mean== | ||

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+ | ==See also== | ||

+ | * [[EBook#Chapter_III:_Probability |Probability Chapter]] | ||

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+ | ==References== | ||

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+ | <hr> | ||

+ | * SOCR Home page: http://www.socr.ucla.edu | ||

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+ | {{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Bayesian_Hierarchical}} |

## Revision as of 23:28, 22 October 2009

## Contents |

## 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 θ_{i} 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.

## Idea of a Hierarchical Model

## Hierarchical Normal Model

## Stein Estimator

## Bayesian analysis for unknown overall mean

## See also

## References

- SOCR Home page: http://www.socr.ucla.edu

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