Gaussian Process
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Suppose $x \sim N(\mu, \Sigma)$, here $\Sigma = \Sigma^T$ and $\Sigma > 0$.
Suppose $x \sim N(\mu, \Sigma)$, and
Let's look at the component $x_1 = \begin{bmatrix} I & 0 \end{bmatrix} x$
Suppose $x \sim N(\mu_x, \Sigma_x)$. Consider the linear function of $x$
Suppose $x \sim N(0, \Sigma)$, and let $c \in \mathbb{R}^n$ be a unit vector
Let $y = c^T x$
$$\mathbf{E}\left(y^2\right) = \lambda_{\text{max}}$$
Suppose $x \sim N(\mu, \Sigma)$, and
Suppose we measure $x_2 = y$. We would like to find the conditional pdf of $x_1$ given $x_2 = y$
By the completion of squares formula
If $x \sim N(0, \Sigma)$, then the conditional pdf of $x_1$ given $x_2 = y$ is Gaussian
$\quad \;$ It is a linear function of $y$.
$\quad \;$ It is not a function of $y$. Instead, it is constant.
$\quad \;$ Conditional confidence intervals are narrower. i.e., measuring $x_2$ gives information about $x_1$