von Mises

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The von Mises distribution is a natural distribution for directions in R2, circular attributes, e.g., angle, time of day, day of the year, phase of the moon, etc.. It is a special case, where D = 2, of the von Mises - Fisher (vMF) distribution over directions in RD.

"The von Mises distribution M(μ,κ) has a mean direction μ and concentration parameter κ. For small κ it tends to a uniform distribution and for large κ it tends to a Normal Distribution with variance 1/κ." -- T. Edgoose, L. Allison & D. L. Dowe, An MML Classification of Protein Sequences that knows about angles and sequences. Pacific Symp. Biocomputing 98, pp.585-596, Jan. 1998.

Probability density function:
f(x | μ, κ) = (1/(2 π I0(κ))) eκ.cos(x-μ)
Using a uniform prior on μ over [0, 2.π)
and prior h3(κ) = κ/(1+κ2)3/2, then
the Fisher information is
F(μ, κ)
= N κ A(κ) N {1 - A(κ)/κ - (A(κ))2}
= N2 κ A(κ) {1 - A(κ)/κ - (A(κ))2}
where I1(κ) = ∂/∂κ I0(κ)
and A(κ) = ∂/∂κ log(I0(κ)) = I1(κ)/I0(κ).
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