Consider a dataset, $\mathcal{D} = (X,Y)$ consisting of points in input space and ground truth values. The regression model; $y_i = f(x_i) + \varepsilon_i$ where $\varepsilon_i \sim \mathcal{N}(0, \sigma^2)$ is a GP regression model if we endow a $0$-mean $\mathcal{GP}$ as prior distribution over the entire function space $f(\cdot)$'s i.e. $$f(\cdot) \sim \mathcal{GP}(0, k(\cdot, \cdot))$$ and condition it on the training data $\mathcal{D}$ to model the joint distribution $f(X)$ and its prediction at test points $f(X')$.
It can be shown that the predictive posterior distribution is given by $\mathcal{N}(\boldsymbol{\mu}, \boldsymbol{K})$ where;
$\boldsymbol{\mu} = K(x',x)K^{-1}(x,x)y = K_{21}K_{11}^{-1}y$
$\boldsymbol{K} = K(x',x') - K(x',x)K^{-1}(x,x)K^T(x', x) = K_{22} - K_{21}K_{11}^{-1}K_{12}$
Now, if I have a new points $(X_n, Y_n)$, how can I use the previously computed $\boldsymbol{\mu}$ and $\boldsymbol{K}$ as prior to the new predictive distribution $\mathcal{N}(\boldsymbol{\mu}_n, \boldsymbol{K}_n)$?
Assuming, that the dimensions match....could I do the following?
$\boldsymbol{\mu}_n = \bigg(K(x'_n,x_n)+K_{21}\bigg)\bigg(K(x_n,x_n)+ K_{11}\bigg)^{-1}y_n$
$\boldsymbol{K}_n = \bigg(K(x'_n,x'_n) + K_{22}\bigg) - \bigg(K(x'_n,x_n)+K_{21}\bigg)\bigg(K(x_n,x_n)+ K_{11}\bigg)^{-1}\bigg(K(x'_n,x_n)+K_{21}\bigg)^T$