[jpeanuts]

The reason those are all "f"s is that they are all versions of the same thing: the function mapping features to outputs, or approximations of it. The capital "F"s refer to random variables/processes describing the same function (using capitals for RVs and lower-case for samples is standard practice in statistics).

By using this notation he is drawing careful distinctions between the various approximations he's using. I think it's pretty good writing.

[psv1]

It's great that's it's consistent. My problem is that the notation only makes sense if you already understand the very thing that he's trying to explain with this notation.

[tel]

That's not entirely true. The point being made is about the consequences of the design being set up with that notation. That design and that notation is reasonably general. It requires some familiarity with notation around mathematical statistics, modeling, Bayesian formalisms, and random variables.

The thing he's trying to explain is how those things interact and what their behavior is.