# 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.