AP Statistics Curriculum 2007 Bayesian Hierarchical

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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.
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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 to 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.
==Idea of a Hierarchical Model==
==Idea of 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==
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* [[EBook#Chapter_III:_Probability |Probability Chapter]]
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==References==
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<hr>
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* SOCR Home page: http://www.socr.ucla.edu
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Current revision as of 21:08, 28 June 2010

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 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 to 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




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