3blue1brown

A draft video on proving the central limit theorem (which I don't particularly love)

Added 2023-08-18 17:15:56 +0000 UTC

Hey folks,

Last month I thought I'd put together a "quick" video offering a sketch for a proof of the central limit theorem. I edited it together most of the way, with several to-do stubs for animations I could throw in to help clarify. When I stepped back, though, I felt like the underlying structure could use some improvement beyond a few clarifying animations.

I'm sharing this partly in the spirit of sharing more partial work before putting projects on a shelf, but also to get your reaction and see if you agree with me on aspects that need improving.

One aim here was to try something stylistically different, beginning with a simple overhead shot of a pen-to-paper line of reasoning. I enjoyed playing around with the style, and it's worth doing more to have simple pen-and-paper segments in videos to convey better what doing math actually feels like. In the end, though, it's not actually that much faster than animating, if at all.

My current beef with it is less about style than substance, though. Here are a few notes I jotted down when I stepped back and tried to view it as if I were a new learner.

* Too many points feel unmotivated, which makes it hard to hold all the different objects in one's head. Most notably, where does the Moment Generating Function come from? In the sea of new terms, it's easy to get a bit lost without a sense of what each one is trying to accomplish.

* A key question is whether knowing that the MGF converges to a single universal shape is enough to justify that the original distribution also converges to a single universal shape. I mention this, but it feels glossed over. I don't even address the sense of convergence that the CLT claims. Really, the argument focussing on MGFs and cumulants is more intuition than proof, given that it's only a subset of distributions that have well-defined MGFs. It could be clearer that most of the video is providing scaffolding that characteristic functions can rigorously fill.

* Arguably the most important object referenced here is the characteristic function, but it's brought up almost as an afterthought. Perhaps a better structure would be to put that function front-and-center and make it as much a video about how probability theorists use Fourier Transforms as it is about anything else.

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I toyed with a second version that took more of a discovery fiction approach. That is, trying to walk down a path for how you might have invented the characteristic function yourself in an attempt to prove the CLT. There's something to that approach, but by this point, I felt I'd overthought the whole thing, and in the grander scheme of things have probably spent way more time on central limit theorem-related videos than is justified.

So, I put this on a shelf, and for August I've turned to a completely separate topic about physics. Stay tuned for updates there, it's shaping up to be a fun one.

In the meantime, if you have thoughts about what you'd like to see whenever I _do_ turn back to add a video to the probability sequence about characteristic functions and proving the CLT, I'm all ears!

-Grant