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I'm trying to work out the expectations of a few functions under the von Mises distribution:

$ p(\theta \mid \mu, \kappa) = \frac{1}{2\pi I_0(\kappa)} \exp\left\{ \kappa \cos \left( \theta - \mu \right) \right\} $

Specifically, for known $\mu$ and $\kappa$,

$ \mathbb{E} \left( \cos\tfrac{\theta}{2} \right) = \frac{1}{2\pi I_0(\kappa)} \int_0^{2\pi} \cos\tfrac{\theta}{2} \exp\left\{ \kappa \cos \left( \theta - \mu \right) \right\} d\theta = ? $

However, my bag of tricks for integrals is fairly small, and I've hit a kind of brick wall.

Ultimately, I'm looking for expressions for $\mathbb{E}(\cos\theta/2)$, $\mathbb{E}(\sin\theta/2)$, $\mathbb{E}(\cos^2\theta/2)$, and $\mathbb{E}(\sin^2\theta/2)$ for a given $\mu$,$\kappa$.

Is there a good lit. source (or trick) for integrals of this type?

Thanks!

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  • $\begingroup$ Most of these integral a will be related to Bessel function derivatives. You can standardize your integrals by using half angle formulas and then thinking about writing them as derivatives in $\kappa$ of Bessel functions. A good place to look is an integral handbook: people.math.sfu.ca/~cbm/aands/page_376.htm $\endgroup$ – Alex R. Jul 15 '16 at 14:57
  • $\begingroup$ Also see the moment calculations here: en.m.wikipedia.org/wiki/Von_Mises_distribution $\endgroup$ – Alex R. Jul 15 '16 at 15:00
  • $\begingroup$ Thanks @AlexR. Since posting, I've been able to get a bit further by shifting the integration limits by $-\mu$, then using trig idents to break apart the leading cosine. In particular, this seems to lead to modified Bessel functions in the $\cos^2$ and $\sin^2$ cases. Yet to crack the first two though... $\endgroup$ – Brad Jul 16 '16 at 16:45
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Ok, after a weekend of wine, cheese and brain-bashing, I solved the integrals. For the sake of posterity:

$ \left\langle \cos(\theta/2) \right\rangle = \frac{1}{\sqrt{2 \pi \kappa} I_0(\kappa)} \left[ e^{ \kappa} \cos(\mu/2) \; \mathrm{erf} \left( \sqrt{2 \kappa} \sin(\mu/2) \right) - e^{-\kappa} \sin(\mu/2) \; \mathrm{erfi} \left( \sqrt{2 \kappa} \cos(\mu/2) \right) \right] $

$ \left\langle \sin(\theta/2) \right\rangle = \frac{1}{\sqrt{2 \pi \kappa} I_0(\kappa)} \left[ e^{ \kappa} \sin(\mu/2) \; \mathrm{erf} \left( \sqrt{2 \kappa} \sin(\mu/2) \right) + e^{-\kappa} \cos(\mu/2) \; \mathrm{erfi} \left( \sqrt{2 \kappa} \cos(\mu/2) \right) \right] $

$ \left\langle \cos^2(\theta/2) \right\rangle = \frac{1}{2} \left[ 1 + \frac{I_1(\kappa)}{I_0(\kappa)} \cos \mu \right] $

$ \left\langle \sin^2(\theta/2) \right\rangle = \frac{1}{2} \left[ 1 - \frac{I_1(\kappa)}{I_0(\kappa)} \cos \mu \right] $

And one more, because why not...

$ \left\langle \cos(\theta/2) \sin(\theta/2) \right\rangle = \frac{1}{2} \frac{I_1(\kappa)}{I_0(\kappa)} \sin \mu $

The one trick I hadn't tried prior to posting was the ticket: let $x = \theta - \mu$ and then use trig identities to simplify the expressions into known integrals.

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