I have a certain confusion. I refer to this paper.

Let's say I have $p$ variables $x_1, x_2, \dots, x_p$ which follow a multivariate Gaussian distribution. Now suppose I have $N$ examples or samples from this distribution. However, some of the values in each example are missing.

Suppose I want to impute the values.

Let's say using only the available examples I calculate the $\mu$ and $\Sigma$ vector which are the mean and covariances (initial estimate). Let $K = \Sigma^{-1}$.

Now I found this formula

$E[x_{ij}|x_{obs,i}, \mu,K]$ = $x_{ij}$ if $x_{ij}$ is observed

$=c_j$ if $x_{ij}$ is not observed

where $c = \mu-(K_{mis,mis})^{-1}*K_{mis,obs}(x_{obs,i}-\mu_{obs})$

$E[x_{ij}x_{ij'}|x_{obs,i},\mu,K]$ = $(K_{mis,mis})^{-1}_{jj'} + c_jc_{j'}$ if both $x_{ij}$ and $x_{ij'}$ are unobserved.

However, $E[x_{ij}x_{ij'}|x_{obs,i},\mu,K]$ = $x_{ij}x_{ij'}$ if both $x_{ij}$ and $x_{ij'}$ are observed.

I didn't get how the term $(K_{mis,mis})^{-1}_{jj'}$ disappears if either $x_{ij}$ or $x_{ij'}$ is observed.


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