# von Mises-Fisher Distribution

The von Mises Fisher Distribution is a multivariate distribution on a hyper sphere. I have decided to share the expectation and covariance of the vMF distribution. The Wikipedia page doesn’t give much info of this distribution.

## Expectation of vMF distribution

Let $C$ be the normalising constant.

Let $\mathbf{y}=\kappa\mathbf{\mu}$. Therefore $\kappa=\sqrt{\mathbf{y}^T\mathbf{y}}$.

This is an interesting result because its saying that the mean of a von Mises-Fisher distribution is NOT $\mathbf{\mu}$. It is infact multiplied a constant $\frac{I_{d/2}(\kappa)}{I_{d/2-1}(\kappa)}$ which is between $(0,1)$. If you think about a uniformly distributed vMF this makes sense ($\kappa\to 0$). If we average all those vectors pointing in different directions it averages very close to 0. This whole ‘averaging’ of unit vectors is what makes the expected value not equal $\mathbf{\mu}$ but a vector pointing in the same direction but smaller in length.

##Covariance of von Mises-Fisher Distribution

Using the same differential approach we can find $E(\mathbf{xx}^T)$ and hence the covariance by using the identity $cov(\mathbf{x},\mathbf{x})=E(\mathbf{xx}^T)-E(\mathbf{x})E(\mathbf{x})^T$. Hence the covariance is,

where $h(\kappa)=\frac{I_{\nu+1}(\kappa)}{I_{\nu}(\kappa)}$ and $\nu=d/2-1$.