# 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?

• Do you mean the variance as in the expected squared deviation from the mean: $1 - I_1(\kappa)/I_0(\kappa)$, or do you mean the dispersion parameter $\kappa$? (See the Wikipedia page for notation if unclear: en.wikipedia.org/wiki/Von_Mises_distribution ) Feb 8 at 0:02
• @jbowman Apologies. I've updated the question to be more accurate. I need a conjugate prior for Kappa. Feb 17 at 19:21

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.

• What is $Z$? What are $a$ and $b$? What is $f$? It can't be arbitrary, for then you would merely be stating (in an overly complicated way) that any distribution with positive support can be a conjugate prior.
– whuber
Feb 17 at 18:34
• $a$ 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 $b$. Feb 17 at 18:43
• You seem to describe any (two-parameter) prior, not a conjugate prior.
– whuber
Feb 17 at 19:21
• From Wikipedia: if the posterior distribution $p(\theta \mid x)$ is in the same probability distribution family as the prior probability distribution $p(\theta )$, the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood function $p(x\mid \theta )$. Feb 18 at 9:01
• Could you please demonstrate that the posterior for such an arbitrary prior is always a von Mises distribution?
– whuber
Feb 18 at 14:05