# Inverse of a sub-matrix

I have a multivariate Gaussian distribution with known $\mu$ and $\Sigma$. I want to evaluate it given a vector $x$. However, some of the attributes of this vector may be unknown, in which case I want to marginalize those values out.

This effectively reduces it to another MV Gaussian distribution with parameters $\hat{\mu}$ and $\hat{\Sigma}$ that are obtained after dropping the rows (and columns) corresponding to unknown attributes in $x$. (Please correct me if I am wrong).

For efficiency of implementation, I pre-compute the inverse of $\Sigma$ and determinant ($|\Sigma|$) to be used while evaluating the distribution. However, when $x$ has unknown values, is there a simple way of determining the inverse of $\hat{\Sigma}$ from the pre-computed inverse of $\Sigma$? Similarly, can I find the determinant easily for $\hat{\Sigma}$?

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If unsuccessful here, try stats.stackexchange.com –  user53153 Feb 12 at 22:11