NotesNov22 - The Bayes Estimators Suppose that X1 X2 Xn is...

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The Bayes Estimators Suppose that X 1 , X 2 , . . . , X n is a sample from a distribution with an unknown parameter θ . Bayes estimators are based on the idea of viewing θ as being a particular value of a random variable. We postulate a prior distribution on θ : a prior density if θ is viewed as continu- ous random variable; a prior pmf if θ can take values in a discrete set.
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Then we use the Bayes formula to compute the posterior density of θ . The observations X 1 , X 2 , . . . , X n can be dis- crete or continuous, not necessarily of the same kind as θ . Discrete observations Suppose that X 1 , X 2 , . . . , X n are iid discrete ran- dom variables with a pmf p X ( x i ; θ ).
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Case 1 : the unknown parameter θ can take only discrete values θ 1 , . . . , θ k . We view θ as a value of a discrete random variable, Θ, with a prior pmf p . Then (Θ , X 1 , X 2 , . . . , X n ) is an ( n +1)-dimensional discrete random vector. By the Bayes formula: P Θ = θ j | X 1 = x 1 , . . . , X n = x n = p ( θ j ) Q n i =1 p X ( x i ; θ j ) P ( X 1 = x 1 , . . . , X n = x n ) .
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This is the conditional (
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