# Are the kernel parameters in Gaussian Processes considered Hyperparameters

In Gaussian Processes, we have kernel parameters. For example, for RBF kernel, we have $$k(x, x')= \sigma_f^2 \exp(-\frac1{2l^2}(x-x')^2)$$, where the $$\sigma_f, l$$ are the two parameters.

My questions are:

1. Are they considered hyperparameters or parameters? I feel they are because they seem not attached to the model learning the data, but I cannot articulate it very well.
2. If they are hyperparameters, then why do I see that their values are often estimated through maximizing the marginal likelihood. For example, in Kevin Murphy's Machine Learning book, or in the fit() function of scikit-learn.GaussianProcessRegressor here. Aren't hyperparameters not learnable and need to be found through grid search or random search?
• Kernels are estimated from data all the time. Whether to call them "hyperparameters" is a matter of preference. Hyperparameters in the Bayesian sense are also estimated from data, oftentimes. Jan 26 at 3:11
• Why are kernel parameters for SVM seem not estimated from data? Also, kernel bandwidth for kernel smoothing/density estimation seems not estimated from data either. Jan 26 at 4:12

In the RBF kernel example you provide, $$\rho_f$$ and $$l$$ are parameters of the kernel, and hyperparameters of any Gaussian process which uses that particular kernel as a parameter.