The problem with the Gaussian distribution is that the normalization constant is too complicated.
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
So, choosing , and defining , we can instead write a Gaussian in the form
By changing , 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 is now a vector)
where is the number of dimensions (Matrix Cookbook). This means that if we are again going to define the constant to try to make the normalization constant disappear, would have to depend on the dimensions of the problem. That seems odd.