Gaussians, Gasses, and Spices.

Today I learned:
1) High-dimensional multivariate gaussians have almost all of their probability in a thin hypersphere shell of standard deviation 1. That’s not as complex as it sounds, so let me try to break it down.

Gaussian: A normal distribution. A bell curve. Basically a function (probability distribution) that looks like a hill, with each of the tails dropping off quite quickly.

Multivariate Gaussian: A normal distribution in more than one dimension. In 2D, that’s a circular hill with a big bulge at the center. In 3D that’s… a spherical blob with a bunch of mass at the center, I guess? A higher-dimensional version is… the same, but in more dimensions!

“Almost all of their probability”: By this, I’m talking about what a statistician might call a “typical set” — that is, there’s some relatively small subregion of the distribution that contains almost all of the mass/probability/weight of the function.

hypersphere shell of standard deviation 1: In the case of a high-dimensional multivariate Gaussian, the typical set is a hypersphere around the center with radius 1 standard deviation — that means that if you take a random sample from, say, a 10,000 dimensional gaussian (this is equivalent to taking 10,000 independent random samples from 10,000 gaussians) and you calculate the distance from the very center of the gaussian to the sample you just took, that distance will almost always be very close to 1. Weird, eh?

Edit: To demonstrate #1, I've put together a quick script, which you can find here: https://github.com/sclamons/QuickDemos (it's MultivariateGaussianTypicalSet.py). Usage:

python MultivariateGaussianTypicalSet.py n_samples n_dimensions

where "n_samples" and "n_dimensions" are the number of samples and the number of dimensions per sample, respectively. Written and tested for Python 2.7. If you have matplotlib installed, it will display a nice histogram of the sample radii; otherwise it will just print the mean radius.

2) There were international treaties in place before World War I banning the use of various kinds of chemical warfare, including “asphyxiating gasses” and various kinds of poisoned weapons, including the use of bullets or artillery containing toxic chemicals. I had always assumed that chemical weapons were *banned* after the war in response to wartime atrocities. Apparently someone foresaw it.

3) It’s harder to tell spices apart by smell than I thought it would be.