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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?

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  • $\begingroup$ 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 ) $\endgroup$
    – jbowman
    Commented Feb 8, 2023 at 0:02
  • $\begingroup$ @jbowman Apologies. I've updated the question to be more accurate. I need a conjugate prior for Kappa. $\endgroup$
    – fountain3
    Commented Feb 17, 2023 at 19:21

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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.

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    $\begingroup$ 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. $\endgroup$
    – whuber ♦
    Commented Feb 17, 2023 at 18:34
  • $\begingroup$ $a$ 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 $b$. $\endgroup$
    – Tom Minka
    Commented Feb 17, 2023 at 18:43
  • $\begingroup$ You seem to describe any (two-parameter) prior, not a conjugate prior. $\endgroup$
    – whuber ♦
    Commented Feb 17, 2023 at 19:21
  • $\begingroup$ 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 $p(x\mid \theta )$. $\endgroup$
    – Tom Minka
    Commented Feb 18, 2023 at 9:01
  • $\begingroup$ Could you please demonstrate that the posterior for such an arbitrary prior is always a von Mises distribution? $\endgroup$
    – whuber ♦
    Commented Feb 18, 2023 at 14:05

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