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I read a book about statistics and machine learning, and can't understand assertion that: let $P(y) \sim N(y|\mu^*, \Sigma^*)$, i.e. multivariate normal distribution $p(y_1|\mu, \Sigma)$, where $\mu_1 = \Sigma(\Sigma_{11}^{-1}\mu_1 - \Sigma_{12}^{-1}(y_2 - \mu_2))$ and $\Sigma = \Sigma_{11} - \Sigma_{12}\Sigma_{22}^{-1}\Sigma_{21}$.

These formulas look difficult and I can't understand how these formulas were obtained. Can anyone give me a hint, how can this be proved?

Maxim
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  • Although your post is vague (what are "$\mu^$" and "$\Sigma^$"?), the formulas suggest you are looking for information about conditional multivariate Normal distributions. The duplicate does the algebra. If you would like some intuition, then see https://stats.stackexchange.com/questions/71260. – whuber Aug 06 '23 at 17:56

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