I have a data matrix $X \in \mathbb{R}^{m \times 4}$, where $m$ is any number of rows, whose data follow a multivariate normal (MVN) distribution.
Suppose that for a given row $i$, the data for the 4th column is missing. We can use the conditional expectation formula for the MVN distribution to estimate the missing value. First, let's partition our data into two parts: $Y_1 = \{X_1, X_2, X_3\}, Y_2 = \{X_4\}$, where $X_i$ represents the column $i$ from matrix $X$. The conditional probability of $Y_2$ given $Y_1$ is expressed as:
$$\mathbb{E}[Y_2 | Y_1 = \mathbf{y_1}] = \mu_2 + \Sigma_{12} \ \Sigma_{11}^{-1} \ (\mathbf{y_1} - \mathbf{\mu_1})$$
The mean vector, $\mathbf{\mu}$, for all four columns is $\mathbf{\mu} = [40, 34, 42, 41]$. The covariance matrix for the data is
$\Sigma = \begin{bmatrix}20 & 7 & 15 & 11\\ 7 &19 &9 & 8\\ 15& 9 &28 & 18 \\ 11 &8 & 18 & 21\end{bmatrix}$
Question: How do I properly partition my matrix $\Sigma$ into the matrices the conditional expectation requires?
I thought $\Sigma_{11}$ was the submatrix that represents the variance and covariance among the non-missing data, $Y_1$, and $\Sigma_{22}$ represents the covariance of among missing ($Y_2$) and non-missing data. So,
$$\Sigma_{11} = \begin{bmatrix} 20 &7 & 15 \\ 7 & 19& 9 \\ 15 & 9 & 28\end{bmatrix}$$
$$\Sigma_{12} = \begin{bmatrix} 11 \\ 8 \\ 18\end{bmatrix}$$
However, with this interpretation, I could write:
$$\mathbb{E}[Y_2 | Y_1 = \mathbf{y_1}] = 41 + \begin{bmatrix} 11 \\ 8 \\ 18\end{bmatrix} \ \begin{bmatrix} 0.085 & -0.011 & -0.42 \\ -0.011 & 00063 & -0.014 \\ -0.042 & -0.014 & 0.060\end{bmatrix} \ \begin{bmatrix} y_1^{(1)} - 40 \\ y_2^{(1)} - 34 \\ y_3^{(1)} - 42 \end{bmatrix} $$
However the dimensions for the product $\Sigma_{12} \ \Sigma_{11}^{-1}$ are off. I could "simply" take the transpose of the $\Sigma_{12}$, but I don't think that'd be correct. I feel like my struggle lies in the poor understanding of this partition.
