Suppose I have the following regression setting:

$y_n = f_n+ \epsilon$ where $f_n = f(x_n)$ and $\epsilon \sim N(0, \sigma^2)$

Let $\textbf{f} = [f(x_1), \ldots , f(x_N)]$, and $\textbf{y} = [y_1, \ldots , y_N]$

We assume $f \sim GP(0, k)$ so that $\textbf{f} \sim N(0, K)$ where $K$ is the $N\times N$ kernel matrix evaluated with points $x_1, \ldots x_N$ using kernel function $k$ of GP prior.

I am interested in calculating the posterior, $p(\textbf{f} | \textbf{y})$

This post gives the following expression for posterior: $p(\textbf{f} | \textbf{y}) \sim N\left(\sigma^{-2}\left( K^{-1} + \sigma^{-2}I\right)^{-1}\textbf{y}, (K^{-1} + \sigma^{-2}I)^{-1}\right)$ and I have personally gone through the derivation of this and found it correct.

However, equation 5 of this post gives the posterior as $p(\textbf{f} | \textbf{y}) \sim N(K(\sigma^2I + K)^{-1}\textbf{y}, \sigma^2 (\sigma^2I + K)^{-1}K)$. I have no idea how they derived this.

So which among these two is actually correct?


I'll assume that the kernel $K$ is invertible and I'll start from the later expression and derive the former.




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