What is the conjugate prior for the Von Mises distribution's precision Does the Von Mises distribution have a conjugate prior for its precision/variance?
Update:
The concentration parameter $\kappa$ (Kappa) seems to control the variance of the Von Mises distribution. If $\kappa = 0$, the distribution is uniform, and for $\kappa > 0$ the distribution becomes more concentrated around its mean. What would be a conjugate prior for Kappa?

A conjugate prior for the concentration parameter $\kappa$ can be any distribution of the form
$$
p(\kappa | a,b) = \frac{\exp(b \kappa)}{I_0(\kappa)^a} \frac{f(\kappa)}{Z(a,b)}
$$
Here $a$ and $b$ are the parameters of the prior. $Z$ is the normalizer. $f$ can be any positive function that does not depend on $a$ and $b$.  Multiplying by a von Mises likelihood function only changes $a$ and $b$, therefore it is a conjugate prior.  Also see Finding the Location of a Signal: A Bayesian Analysis.
