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Mean-field variance

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A normal posterior over $x$ and $y$ has $Var (x) = 9$ , $Var (y) = 4$ and $Cov (x, y) = 3$ . The best independent normal approximation, the one that minimises $D_{KL} (q ∥ p)$ , gives $x$ the conditional variance of $x$ given $y$ . What variance does it give $x$ ? Answer as a fraction in lowest terms.

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Hint

Take the true variance of $x$ and subtract what $y$ explains.

Solution

The part explained by $y$ is $\frac{3 ^{2}}{4} = \frac{9}{4}$ .
The approximation gives $9 - \frac{9}{4} = \frac{27}{4}$ , so the answer is $\frac{27}{4}$ .

Answer: $\frac{27}{4}$

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