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I am having trouble showing that [itex]\mathbb{E}\left[(Y-\mathbb{E}[Y|X])^{2}\right]=0[/itex].

I understand the proof of why E[Y|X] minimizes the mean square error, but I cannot understand why it is then equal to zero.

I tried multiplying out the square to get [itex]\mathbb{E}\left[Y^{2}\right]-2\mathbb{E}\left[Y\mathbb{E}[Y|X]\right]+\mathbb{E}\left[\mathbb{E}[Y|X]\mathbb{E}[Y|X]\right][/itex]

but have not been able to justify [itex]\mathbb{E}\left[Y\mathbb{E}[Y|X]\right]=\mathbb{E}\left[Y^{2}\right]

[/itex] or [itex]\mathbb{E}\left[\mathbb{E}[Y|X]\mathbb{E}[Y|X]\right]=\mathbb{E}\left[Y^{2}\right][/itex].

Thanks in advance.

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# Why does conditional probability used in mean square error equal zero?

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