von Mises–Fisher distribution

Remarks

Example: for $p = 3$, $I_{1/2}(\kappa) = \sqrt{2/(\pi \kappa)} \sinh \kappa$, so $C_3(\kappa) = \kappa / (4 \pi \sinh \kappa)$. Check: in spherical coordinates with $\mu$ as north pole, $\int_{S^2} e^{\kappa \cos \theta} \, d\sigma = 2\pi \int_0^\pi e^{\kappa \cos \theta} \sin \theta \, d\theta = 4 \pi \sinh(\kappa)/\kappa$. Non-example: a density proportional to $\exp(\kappa (\mu^T x)^2)$ with $\kappa > 0$ is not von Mises–Fisher. It takes the same value at $x$ and $-x$, so its maximum is attained at both $\mu$ and $-\mu$, while a von Mises–Fisher density has a single maximum. The density is largest at $x = \mu$ and smallest at $x = -\mu$, and it depends on $x$ only through the angle to $\mu$. For $p = 2$, writing $x = (\cos \theta, \sin \theta)$ gives the von Mises density $e^{\kappa \cos(\theta - \theta_0)} / (2 \pi I_0(\kappa))$. The maximum likelihood estimate of $\mu$ from a sample is its mean direction.