law of iterated expectations

law of iterated expectations - Notes on Law of Iterated...

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Notes on Law of Iterated Expectations Cyrus Samii The law of iterated expectations is the foundation of many derivations and theorems in applied statistics. I provide the basic statement of the law and then illustrate how it applies to some important results. A basic statement is as follows: E ( Y ) = E (E ( Y | X )) . Here is a proof for the continuous case: E (E ( Y | X )) = Z E ( Y | X ) f X ( x ) dx = Z Z yf Y | X ( y | x ) dy f X ( x ) dx = Z Z yf X,Y ( x, y ) dxdy = Z y Z f X,Y ( x, y ) dx dy Z yf Y ( y ) dy = E ( Y ) , and for the discrete case, E (E ( Y | X )) = X x E ( Y | X = x ) p ( x ) = X x ˆ X y yp ( y | x ) ! p ( x ) = X x X y yp ( x, y ) = X y y X x p ( x, y ) = X y yp ( y ) = E ( Y ) . Example 1. Suppose Y | X N (2 X, 1) and X Multinomial ( . 25 , . 25 , . 5) . This defines a joint distribution for Y and X (since f ( Y | X ) f ( X ) = f ( X, Y ) ). Take a large number of draws from this joint distribution. Average the Y realizations. Then, bin the Y realizations by their associated X values, take the average of the Y realizations within each bin, and finally take the probability-weighted average over the bins. This should equal the mean of the Y s. Here is a demonstration with a finite (though rather large) sample using R: > n <- 5000 > p <- c(.25,.25,.5) > x <- t(rmultinom(n,1,p))% * %c(1,2,3) > y <- rnorm(n, mean=2 * x, sd=1) > mean(y) [1] 4.500761 > tapply(y, x, mean) 1 2 3 1.994880 3.973878 6.033394 > tapply(y, x, mean)% * %as.matrix(p) [,1] [1,] 4.508887 As the example illustrates, E ( Y | X ) is itself a random variable. In the example, we have, E ( Y | X ) = 2 if X = 1 4 if X = 2 6 if X = 3 , 1
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where the probability of any of those E ( Y | X ) realizations is equal to the probability of realizing X = 1 vs X = 2 vs X = 3 , respectively. Thus, E ( Y | X ) is random with respect to the variation in X , and so the outer expectation is taken with respect to this source of random variation—namely, the density of X . Iterated expectations can be taken with nested conditioning sets. For example, we have E ( Y | X ) = E (E ( Y | X, Z ) | X ) . The proof is no more complicated than what we have above—just substitute in the appropriate conditional densities (e.g., replace f Y with f Y | X ). What is important is the statistical interpretation, which typically looks at the condition-
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