Today I Learned

Some of the things I've learned every day since Oct 10, 2016

38: Orthogonality Property of Conditional Expectation

The orthogonality property of conditional expectation states that, where X, Y are random variables, \textrm{E}[Y|X] is the conditional expectation of Y dependent on X, and \phi (X) is any function of X,

\textrm{E}(Y - \textrm{E}[Y|X]) \phi(X)) = 0.

This can be interpreted as meaning that the ‘vector’/function Y - \textrm{E}[Y|X] is orthogonal to the space of functions of X, which implies that \textrm{E}[Y|X] gives the minimum mean squared error of Y given X.

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