The multi-variate Gaussian (16.3) can be thought of as one member of a class of PDFs

$\displaystyle \log p({\bf x}) = - \frac{1}{2} \left[\left({\bf x}- \mbox{\boldm...
...; {\bf C}^{-1} \; \left({\bf x}- \mbox{\boldmath$\mu$}\right) \right]^{p/2} -c,$ (16.11)

where for Gaussian, $ p=2$ and

$\displaystyle c= \frac{N}{2}\log(2\pi) +\frac{1}{2} \log {\rm det}({\bf C}).$

For arbitrary $ p$, we have

$\displaystyle c = -\log(p) + \left(\frac{N}{p}+1\right) \log(2) + \frac{N}{2}\l...
...pi) - \log\Gamma(N/2)
+ \log\Gamma(N/p) +\frac{1}{2} \log {\rm det}({\bf C}).$

For $ p>2$, the distribution tends to a flat-topped elliptical plate in $ N$-space with sharp shoulders. For $ p<2$, the distribution has higher tails. These non-Gaussian distributions may be useful in some applicationw where tail behavior needs to be modified.

Baggenstoss 2017-05-19