A Cleaner Gaussian Distribution

The problem with the Gaussian distribution is that the normalization constant is too complicated.

p(x) = \frac{1}{\sigma\sqrt{2\pi}}\exp(-\frac{x^2}{2\sigma^2})

I admit it really isn’t particularly complicated, but in its many forms– multivariate, conditional, CDF, etc. these things continue to cause annoyance.  In particular, I am frequently finding that I introduce bugs when I write code using Gaussians.

Now, can this be simplified?  It can.  Notice that

\int_{x=-\infty}^\infty a^{-\frac{x^2}{\sigma^2}}=\frac{\sigma\sqrt{\pi}}{ \sqrt{\log(a)} }

So, choosing a=e^\pi, and defining \text{axp}(x)=a^x, we can instead write a Gaussian in the form

p(x)=\frac{1}{\sigma} \text{axp}(-\frac{x^2}{\sigma^2}).

By changing \sigma, this represents any normal Gaussian.

Now, that’s slightly nicer than a regular gaussian, but can it extend to higher dimensions?  (I admit I have to look up the normalization constant for a multivariate Gaussian every time I use one.)  Unfortunately, it doesn’t seem so.  The trouble is that (here x is now a vector)

\int_{x=-\infty}^\infty \exp(-\frac{1}{2}x^T \Sigma^{-1} x)=(2\pi)^{d/2} |\Sigma|^{1/2}

where d is the number of dimensions (Matrix Cookbook).  This means that if we are again going to define the constant a to try to make the normalization constant disappear, a would have to depend on the dimensions of the problem.  That seems odd.

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Quantum Mechanics

This “diavlog” is fascinating.  They discuss the meaning of quantum mechanics, and how that relates to the issue of what a “measurement” is, and how that relates to what “consciousness” is.

I like this site’s trick of speeding up the video by a factor of 1.4, with clever signal processing to avoid making the participants sound like chipmunks.  It is strange that we can effortlessly listen to people speaking faster than they naturally speak.  (On first watching these videos, one doesn’t even notice that it has been sped up– the participants just seem to have had way too much coffee.)