When investigating distributions from a completely different source, I thought for a while
these are all pretty gaussian! except for the bits on the right, and I know where those come from
then recognized that their tails were wider and thought
maybe they’re … student-t or something? [confused]
and eventually realized
oh, of course, this is a mixture of lots of gaussians; each natural class induces a gaussian in the limit, but this distribution represents a whole slew of natural classes, and most of their means are quite similar, so yeah, it’s gonna look like a fatter-tailed gaussian with some “outliers”
and I think this probably happens quite a lot, because while CLT ==> gaussian, nature rarely says “here’s exactly one type of object!”
If you sum enough Gaussians you can get close to any distribution you want. I’m not sure what the information behind “it’s Gaussian” in this context. (It clearly doesn’t look like a mixture of a few Gaussians...)
It clearly doesn’t look like a mixture of a few Gaussians
It does to me. If their means are close enough compared to their variances, it’ll look like a unimodal distribution. For a good example, a student t distribution is a mixture of gaussians with the same mean and variances differing in a certain way, and it looks exactly like these images.
When investigating distributions from a completely different source, I thought for a while
these are all pretty gaussian! except for the bits on the right, and I know where those come from
then recognized that their tails were wider and thought
maybe they’re … student-t or something? [confused]
and eventually realized
oh, of course, this is a mixture of lots of gaussians; each natural class induces a gaussian in the limit, but this distribution represents a whole slew of natural classes, and most of their means are quite similar, so yeah, it’s gonna look like a fatter-tailed gaussian with some “outliers”
and I think this probably happens quite a lot, because while CLT ==> gaussian, nature rarely says “here’s exactly one type of object!”
If you sum enough Gaussians you can get close to any distribution you want. I’m not sure what the information behind “it’s Gaussian” in this context. (It clearly doesn’t look like a mixture of a few Gaussians...)
It does to me. If their means are close enough compared to their variances, it’ll look like a unimodal distribution. For a good example, a student t distribution is a mixture of gaussians with the same mean and variances differing in a certain way, and it looks exactly like these images.
See the first image here: https://en.m.wikipedia.org/wiki/Student’s_t-distribution