I definitely was thinking they were literally the same in every case! I corrected that part and learned something. Thanks!
Maxwell Peterson
Karma: 134
Convolution as smoothing
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?)
Yup, totally! I recently learned about this theorem and it’s what kicked off the train of thought that led to this post.
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?
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.
The central limit theorem in terms of convolutions
Those are good points.
We already tried really really hard to reduce smoking in the US. I think all these curves, where effort is on the x axis and benefit on the y, see decreasing returns once you have already put in a lot of effort.
Another way of putting it: People I know who I advise to distance more and wear a mask more might disagree and argue with me, but they’ll at least consider my arguments and say why they’re right and engage. A person I know who smokes, who I advise to stop, will just laugh and blow me off: “whatever dude”. They’ve heard it before. So among people I know, “hey beware covid” is a way more effective message than “hey beware smoking”, so I barely ever bother with the latter.
Thanks!
That helps—I wasn’t sure whether there might maybe be some small special intuitive difference in Borel or Jordan that could correspond to a different real world example, but now I think that’s definitely a No.
Intuitively, a metric outputs how different two things are, while a measure outputs how big something is.
In terms of inputs and outputs: a metric takes two points as input, and outputs a positive real number. A measure takes one set as input, and outputs a positive real number.
Love it. Never tried the old editor, but had tried writing posts a couple times in the past, on other sites. I’d always get stuck screwing with the editor settings and trying to figure it out, or losing my work, or whatever. This current LW editor makes it so easy that I finally finished a post (and then two more)! The editor isn’t the whole reason for that, but it’s definitely a factor.
Yeah, this makes sense. Hmm. I’ll think about this more then edit the post. Thanks
Ahh. I could very well be wrong. Trying to understand this; visualization-wise, are you saying that instead of visualizing the point moving around, with the green circles fixed, we should be visualizing the green circles moving around, with the point fixed?
Nice recommendation—learned multiple things from it
Fixed!
Thanks—fixed
Examples of Measures
Your comment made me realize that I didn’t actually know what it meant to add random variables! I looked it up and found that, according to Wikipedia, this corresponds (if the RVs are independent) to what my main source (Jaynes) has been talking about in terms of convolutions of probability distributions. So I’m gonna go back and re-read the parts on convolution.
But I still want to go out on a limb here and say thatSo, to say anything useful about a family of random variables, they all have to live on the same space
sounds to me like too strong a statement. Since I can take the AND of just about any two propositions and get a probability, can’t I talk about the chance of a person being 6 feet tall, and about the probability that it is raining in Los Angeles today, even though those event spaces are really different, and therefore their probability spaces are different? And if I can do that, what is special about the addition of random variables that makes it not applicable, in the way AND is applicable?
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.