handout6 - ISyE8843A, Brani Vidakovic 1 1.1 Handout 6...

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ISyE8843A, Brani Vidakovic Handout 6 1 Priors Continued 1.1 Nuisance Parameters Assume that unknown parameter is ( θ 1 2 ) but we are interested only in θ 1 . The parameter θ 2 is called nuisance parameter. How do we handle nuisance parameters? Suppose π ( θ 1 2 ) is the joint prior for ( θ 1 2 ) . The posterior is π ( θ 1 2 | x ) f ( x | θ 1 2 ) · π ( θ 1 2 ) . The marginal posterior of interest is obtained by averaging over the nuisance parameter, π ( θ 1 | x ) = Z π ( θ 1 2 | x ) 2 , or π ( θ 1 | x ) = Z π ( θ 1 | θ 2 ,x ) π ( θ 2 | x ) 2 . Example 1: Let X = ( X 1 ,...,X n ) be a sample from normal N ( μ,σ ) distribution. Assume π ( μ,σ 2 ) = 1 σ 2 . The posterior distribution is (by slight abuse of notation) π ( μ,σ 2 | X ) 1 σ - n - 2 e - 1 2 σ 2 P n i =1 ( X i - μ ) 2 = 1 σ - n - 2 e - 1 2 σ 2 [( n - 1) s 2 + n ( ¯ X - μ ) 2 ] , where s 2 = 1 n - 1 n i =1 ( X i - ¯ X ) 2 . Show that π ( σ 2 | X ) is IG amma ( n - 1 2 , ( n - 1) s 2 2 ) . This distribution is sometimes referred as scaled inverse χ 2 , inv - χ 2 ( n - 1 ,s 2 ) , see Handout 0. The joint posterior can be represented as π ( μ,σ 2 | X ) = π ( μ | σ 2 ,X ) · π ( σ 2 | X ) , π ( μ,σ 2 | X ) 1 σ n +2 e - ( n - 1) s 2 2 σ 2 · p πσ 2 /n p πσ 2 /n e - n/ (2 σ 2 )( μ - ¯ X ) 2 1 σ n +1 e - ( n - 1) s 2 2 σ 2 · N ( ¯ X,σ 2 /n ) 1 ( σ 2 ) n - 1 2 +1 e - ( n - 1) s 2 2 σ 2 · N ( ¯ X,σ 2 /n ) ∝ IG amma ( n - 1 2 , ( n - 1) s 2 2 ) · N ( ¯ X,σ 2 /n ) 1
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Exercise 1: Derive prior and posterior predictive distributions for the model in the Example 1 above. Exercise 2: Consider the Normal Inverse Gamma prior. This prior is conjugate. Find the parameters of the posterior. Exercise 3:
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This note was uploaded on 10/23/2011 for the course ISYE 8843 taught by Professor Vidakovic during the Spring '11 term at Georgia Institute of Technology.

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handout6 - ISyE8843A, Brani Vidakovic 1 1.1 Handout 6...

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