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5102-Lecture-13-Board-Notes

5102-Lecture-13-Board-Notes - NOTES FOR STAT 5102-004...

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NOTES FOR STAT 5102-004 LECTURE 13 16 FEBRUARY 2011 1. Predictive Region for a Normal with known variance Let Y 1 , . . . , Y n +1 iid N( μ, σ 2 ) with σ 2 known. Suppose we observe y = ( y 1 , . . . , y n ), and want to make predictions concerning Y n +1 . Note that Y n +1 - ¯ Y n N(0 , σ 2 (1 + 1 n )) since ¯ Y n and Y n +1 are independent normals with common mean μ . Thus, from r n n + 1 1 σ ( Y n +1 - ¯ Y n ) N(0 , 1) we can use equalities such as Pr ¯ Y n - σ q 1 + 1 n 1 . 96 Y n +1 ¯ Y n + σ q 1 + 1 n 1 . 96 = 0 . 95 to create level γ prediction intervals. Specifically, ¯ y n ± σ q 1 + 1 n 1 . 96 is a level 0 . 95 prediction interval for y n +1 . By comparison, recall that Pr ¯ Y n - σ q 1 n 1 . 96 μ ¯ Y n + σ q 1 n 1 . 96 = 0 . 95 led to a 95% confidence interval for μ . The difference between a confidence interval and a prediction interval is the difference between trying to cover a fixed quantity and trying to cover a random quantity. Prediction always has to account for the sampling variability in a new observation. 2. Posterior for θ with τ known 2.1. Normal Conjugate Prior. Let Y 1 , . . . , Y n iid N( θ, 1 τ ) with τ known. Assume a normal prior for θ with mean μ 0 and precision λ 0 τ . The prior density is π ( θ ) = (2 π ) - 1 2 λ 1 2 0 τ 1 2 exp ( - 1 2 λ 0 τ ( θ - μ 0 ) 2 ) . The likelihood function satisfies lik( θ | y ) τ n 2 exp ( - 1 2 τ n i =1 ( y i - θ ) 2 ) which, letting n i =1 ( y i - ¯ y ) 2 = k (I - P) y k 2 , can be separated out into lik( θ | y ) exp ( - 1 2 ( θ - ¯ x ) 2 ) τ n 2 exp ( - 1 2 τ k (I - P) y k 2 ) . (1) 1
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2.2. Approaching the Posterior. The posterior density is proportional to the prior den- sity times the likelihood: π ( θ | y ) lik( θ | y ) π ( θ | τ ) τ 1 2 exp ( - 1 2 τ λ 0 ( θ - μ 0 ) 2 + n ( θ - ¯ x ) 2 ) (2) × τ n 2 exp ( -
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