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∙ if we define “best” to be minimum mean square

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Unformatted text preview: ∙ If we define “best” to be minimum mean square error then E Y | X is the best we can do. 53 ( ce8 ) (Minimum MSE Predictor) Among all functions of X , the conditional expectation minimizes the mean squared error. To be precise, let x ≡ E Y | X x . We want to use the outcome on X to predict Y , so our prediction functions can be written as g x . But we need to consider the random variable g X because we want a criterion that considers all possible outcomes on X . (For example, if X contains high school GPA and standardized test scores, we need to allow that X takes on many different values in the pool of applicants.) 54 ∙ Define MSE g E Y − g X 2 , where the expected value is over the joint distribution of X , Y . For the conditional mean , MSE E Y − X 2 55 ∙ The formal claim is that, if E Y 2 , then for any function with E g X 2 , E Y − X 2 ≤ E Y − g X 2 . In fact, we can say even more. Write Y − g X 2 Y − X X − g X 2 Y − X 2 2 Y − X X − g X X − g X 2 56 Take the expected value conditional on X : E Y − g X 2 | X E Y − X 2 | X 2 X − g X E Y − X | X X − g X 2 Var Y | X X − g X 2 because E Y − X | X 0 and E Y − X 2 | X Var Y | X . Now take the expectation and use the LIE: E Y − g X 2 E Y − X 2 E X − g X 2 ≥ E Y − X 2 57 2 . 2 . Properties of Conditional Variances and Covariances ( cv1 ) For any function a x , Var a X | X 0. ( cv2 ) For functions a x and b x , Var a X Y b X | X a X 2 Var Y | X ( cv3 ) The unconditional variance of Y can be expressed as Var Y E Var Y | X Var E Y | X ≡ E X Var Y | X Var X E Y | X ≡ E X 2 X Var X X 58 Proof: Let Y E Y and X E Y | X . Then Y 2 E Y − Y 2 ....
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∙ If we define “best” to be minimum mean square error...

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