Maxwell Peterson


The central limit theorems


Convolution as smoothing

Hm. What you're saying sounds reasonable, and is an interesting way to look at it, but then I'm having trouble reconciling it with how widely the central limit theorem applies in practice. Is the difference just that the space of functions is much larger than the space of probability distributions people typically work with? For now I've added an asterisk telling readers to look down here for some caution on the kernel quote.

Convolution as smoothing

Yes, that sounds right - such an  exists. And expressing it in fourier series makes it clear. So the “not much” in “doesn’t much matter” is doing a lot of work.

I took his meaning as something like "reasonably small changes to the distributions  in don’t change the qualitative properties of ". I liked that he pointed it out, because a common version of the CLT stipulates that the random variables must be identically distributed, and I really want readers here to know: No! That isn’t necessary! The distributions can be different (as long as they’re not too different)!

But it sounds like you’re taking it more literally. Hm. Maybe I should edit that part a bit.

Why are young, healthy people eager to take the Covid-19 vaccine?

Your 20% link is the cardiology link repeated. I think I know the link you meant: this Lancet study

(I'd caution that a number of journalists mis-read the abstract and reported that nearly 20% of people had a first-time mental health diagnosis after COVID - that isn't so! Only 5.8% had a first time diagnosis. The near-20% (18.1%) includes people already diagnosed with a mental health condition. You might have known this already but I wasn't sure from your phrasing, and this specific error on this study is common so I thought I'd mention it.)

The central limit theorem in terms of convolutions

Ahh - convolution did remind me of a signal processing course I took a long time ago. I didn't know it was that widespread though.  Nice.

The central limit theorem in terms of convolutions

I definitely was thinking they were literally the same in every case! I corrected that part and learned something. Thanks!

The central limit theorem in terms of convolutions

Ha - after I put the animated graphs in I was thinking, “maybe everyone’s already seen these a bunch of times...”.

As for the three functions all being plotted on the same graph: this is a compact way of showing three functions: f, g, and f * g. You can imagine taking more vertical space, and plotting the blue line f in one plot by itself - then the red line g on its own plot underneath - and finally the black convolution f * g on a third plot. They’ve just been all laid on top of each other here to fit everything into one plot. In my next post I’ll actually have the more explicit split-out-into-three-plots design instead of the overlaid design used here. (Is this what you meant?)

The central limit theorem in terms of convolutions

Yup, totally! I recently learned about this theorem and it’s what kicked off the train of thought that led to this post.

The central limit theorem in terms of convolutions

Gotcha. The non-linearity part “breaking” things makes sense. The main uncertainty in my head right now is whether repeatedly convolving in 2d would require more convolutions to get near gaussian than are required in 1d - like, in dimension m, do you need m times as many distributions; more than m times as many,;or can you use the same amount of convolutions as you would have in 1d? Does convergence get a lot harder as dimension increases, or does nothing special happen?

Have epistemic conditions always been this bad?

I was at the University of Washington from the beginning of 2013 to the end of 2014 and noticed almost none of this. I was in math and computer science courses, and outside of class mostly hung out with international students, so maybe it was always going on right around the corner, or something? But I really don’t remember feeling anything like the described. I took a Drama class and remember people arguing about... Iraq...? for some reason, with there being open disagreement among students about some sort of hot-button topic. More important, one of the TAs once lectured to the whole entire class of a couple hundred students about racism in theater, and at times spoke in sort of harsh “if you disagree, you’re part of the problem” terms... and some students walked out! Walking out is a pretty strong signal, and not the kind of thing you do if you’re afraid of retribution.

This is all an undergraduate perspective. Any effect like this could be a lot stronger among people trying to actually make a career at the school.

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