## Return to Answer

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For example, let us suppose that $a=0, b=N+1$:

• you have a noisy observation $Y=(y_1, y_2)=(O_a, O_b)$ with know covariance matrix $\Sigma_Y$
• the data you are looking for, $X=(z_1, \ldots, z_N) \in \mathbb{R}^N$, have a known covariance matrix $\Sigma_X$
• the covariance matrix $E[X Y^t] = \Sigma_{X,Y}$ is also known.
• A quick way to find the conditional distribution of $X$ knowing $Y$ is to write$$X = AU + BV$$where $U,V$ are independent standard Gaussian random variable of size $2$ and $N$ respectively, while $A \in M_{N,2}(\mathbb{R})$ and $B \in M_{N,N}(\mathbb{R})$ and $C \in M_{2,2}(\mathbb{R})$. Because

• $CC^t = \Sigma_Y$ gives you $C=\Sigma_y^{\frac{1}{2}}$,
• $AC^t = \Sigma_{X,Y}$ then gives you $A=\Sigma_{X,Y}\Sigma_y^{-\frac{1}{2}}$,
• $AA^t + BB^t = \Sigma_{X}$ then gives you $B=(\Sigma_{X}-\Sigma_{X,Y}\Sigma_y^{-1}\Sigma_{X,Y})^{\frac{1}{2}}$,
• the $3$ matrices are easily computable, and $C$ is invertible in the case you are considering. This shows that if you know that $Y=y$, the conditional law of $X | Y=y$ is given by$$X = AC^{-1}y + BV,$$which is a Gaussian vector with mean $AC^{-1}y = \Sigma_{X,Y}\Sigma_y^{-1}y$ and covariance $BB^t = \Sigma_{X}-\Sigma_{X,Y}\Sigma_y^{-1}\Sigma_{X,Y}$

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if you consider a Gaussian vector $V=(X,Y) \in \mathbb{R}^{d=m+n}$, you know how to find the conditional distribution of $X$ knowing the value of $Y=y$, right ? This is exactly the same thing here.