von Mises

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 R^D.

"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))) e^κ.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).