The paragraph beginning with "Let’s consider how we might apply the Bayesian framework..." where he introduces the notation is a great example of everything I hate about mathematical notation. We have big-F, small-f, f-of-x, f-sub-x, f-star, big-F-star... and then he decides to abbreviate what he just introduced. If I didn't know what's happening and I was trying to understand this for the first time, I would have no chance and would just give up right there.
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.
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.
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.
Hey, author here. The writing on this section was a bit tricky to get right, but we did our best to keep it as clean as possible while still being precise about the concepts we were considering. And it's definitely not perfect; I've just made some small edits to hopefully make things a bit more clear.
This blog post is a response to ongoing discussion with the Bayesian community, so it was primarily aimed at a more technical audience. If you have any suggestions for how to make the writing more accessible, without becoming so overly expository that the mathematically-robust folk lose interest, I would love to discuss them.
So here’s one thing about mathematical notation. If it uses similar symbols/scripts/subscripts that means that the objects are related, up to the minor difference expressed. So choosing “f” for everything is often a deliberate and well-motivated decision (Very much like naming variables). Doing that well is an art form.
Unfortunately, yes, it does take some time to get used to it (both reading, and generating such names), but IMHO it’s far better notation than otherwise.
