What is the benefit of a gaussian procces?

I wonder about the advantage of a gp over "standard" regression tools like a loess. A gp fits to the data as best as possible, like every model tries to do this. More or less, let's keep it that simple. Now, all I see is that a gp adds a confidence intervall around it's regression curve which becomes broader the lass data is available. So where is the benefit of a gp?

A similar question has been answered, but it's over my head, at least at the moment. Can someone please explain in simple words?

Some disadvantages of LOWESS vs GP are the following:

• LOWESS will fail even for a moderate input dimension of data, while GP works in this case for right kernel selected
• For LOWESS to work we need a dense training sample of points over the whole design space. For GP the requirements are less strict
• We need additional tricks to control outliers

Some advantages of GP vs LOWESS are the following:

• Both prediction and uncertainty estimate have analytical forms: at each point $$x$$ we have a posterior distribution given the data $$\mathcal{N}(\mu(x), \sigma^2(x))$$ with $$\sigma^2(x)$$ has analytical form and is natural uncertainty estimate
• GP controls over the smoothness of the model by the selection of a proper kernel
• Straightforward estimation of parameters: we write likelihood for the data and maximize it with respect to parameters of the kernel
• GP can be treated as a special problem for the selection of a function from RKHS space
• Theoretical justification: it is a solution of Kolmogorov-Wiener equations, so we minimize L2 error over all possible estimators under the assumption about the Gaussian joint data distribution
• An interpolation without giving up the smoothness of the model, so the predictions $$\hat{y}(x)$$ equal to the true values $$y(x)$$.

IMO the most important points are limitation of LOWESS to moderate dimensions and absence of analytical and principled global prediction and uncertainty estimate. Also, I believe, that in practice GP just works better, than most of the kernel methods including LOWESS.