# How to understand the Gaussian Progress Regression?

I know Gaussian Progress Regression is completed determined by its mean and covariance functions. For given training and test data, I can compute the covariance matrix correctly like a machine, and I can also describe the main procedure of GPR (compute the joint probability over training data and estimation point, cut a piece through the marginal likelihood....etc), but could anyone can describe what is Gaussion Progress for Regression using your OWN words?

I'm far from being an expert, but here I go. The posterior mean function in a GP regression is a linear smoother, and it can be seen as a generalized nearest-neighbors prediction method: for any point $x^*$ in regressor space, a mean predicted output $\hat y^*$ is computed from the training data $(\mathbf{X}, \mathbf{Y})= (x_i, y_i)_{i \in [1, n]}$ as a weighted average:
$\hat y^*(x^*, \mathbf{X}, \mathbf{Y}) = \sum_{i=1}^n w_i(x^*, \mathbf{X}) \cdot y_i$
where the weights $w_i$ will typically be high for training observations "close" to point x$^*$ (in regressor space, ie. using only $\mathbf{X})$, and low for "far away" observations.
Changing the covariance function (kernel) and hyperparameters changes the definition of how close two observations $(x_1, x_2)$ are, and this will change the weights $w_i$. But in the end the mean prediction $\hat y^*$ is always just an average of the observed $y_i$ associated with observed $x_i$ that are close to $x^*$.
Additionally to this mean prediction $\hat y^*$, GP regression gives you the (Gaussian) distribution of $y$ around this mean, which will be different at each query point $x^*$ (in contrast with ordinary linear regression for instance, where only the predicted mean of $y$ changes with $x$ but where its variance is the same at all points).
A fitted GP even gives you the full joint distribution of any vector $(y^*_1, y^*_2, ... y^*_t)$ given the query inputs $(x^*_1, x^*_2, ... x^*_t)$, which is really a lot of information (including estimated partial derivatives $\widehat{\frac{\partial y}{\partial x}}$), but for regression you're probably mostly interested in the predicted mean and variance.