frequentist vs Bayesian approaches to Gaussian Processes? I've been reading this blog post, which has been tremendously helpful in understanding Gaussian Processes (GP.) The author has used the terms "prior", "posterior", and "95% confidence interval" throughout; this made it difficult for me to tell if he was approaching GP from a Frequentist or Bayesian perspective. In equation 11, he defines the log marginal likelihood then uses scipy to optimize the hyperparameters, L and Sigma. He doesn't explicitly call it "MLE" but the concept seems analogous enough for me to consider his approach Frequentist. (Is this valid?)
And if so, how might a Bayesian approach GP? My currently thinking is: Rather than optimize the log marginal likelihood for point estimates of L and Sigma, sample hyperparameter values from the joint distribution. (Is this valid?)
Lastly, the author makes some decisions that made the distinction between paradigms a little blurry for me. Namely, early in the article he discusses the prior, sets arbitrary values for L and Sigma, generates some fake X input data, then uses the Kernel(X,X) to generate a multivariate gaussian (w/ one dimension per data point) and samples from this multivariate gaussian 3 times to show what this "prior over functions might look like". And this has caused me a decent amount of confusion as it sounds quite Bayesian.

Could anyone offer some clarification?
Edit: Wow, there is no agreement on this topic, see extended conversation.
 A: Gaussian-process (GP) regression is almost always handled in a Bayesian manner. When people talk about "picking the right kernel" for their GP, they mean for the prior, and it has a dramatic effect on the predictions they make once updated via data to their posterior. The article you linked makes this idea pretty clear.
The author's use of the term "confidence interval" is a misnomer. They are just plotting the spread as dictated directly by the covariance matrix of the assumed Gaussian distribution. These are actual probabilities over the thing being estimated. I.e. your picture of the author's prior could be described as "$f(x_i)$ under the prior has a 95% probability of being between -2 and 2 $\forall i$."
The "marginal likelihood" maximization in Eq.11 is not the "training" (the "training" is the Bayesian updating of the prior GP to the posterior GP). Rather, it is sort of data-driven tuning of the prior by selecting kernel hyperparameters that maximize the "evidence" (denominator in Bayes rule), which the author calls "marginal likelihood."
