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# handout6 - ISyE8843A Brani Vidakovic 1 1.1 Handout 6 Priors...

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ISyE8843A, Brani Vidakovic Handout 6 1 Priors Continued 1.1 Nuisance Parameters Assume that unknown parameter is ( θ 1 , θ 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 , θ 2 ) is the joint prior for ( θ 1 , θ 2 ) . The posterior is π ( θ 1 , θ 2 | x ) f ( x | θ 1 , θ 2 ) · π ( θ 1 , θ 2 ) . The marginal posterior of interest is obtained by averaging over the nuisance parameter, π ( θ 1 | x ) = Z π ( θ 1 , θ 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: What if the prior is π ( θ, σ 2 ) = 1 σ ? (1) This prior is noninformative independence prior since it is a product of non-informative, translation invariant priors for θ and σ 2 . This prior, although not Jeffreys’ for the problem, was ultimately recommended by Jeffreys in his book from (1961). For the mathematically thirsty, the prior in (1) is the right invariant Haar
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handout6 - ISyE8843A Brani Vidakovic 1 1.1 Handout 6 Priors...

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