von Mises

The von Mises distribution is a "natural" distribution for circular attributes, e.g. angles, time of day, day of the year, phase of the moon, etc.. "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:

f(x | μ, κ) = (1/(2.π.I₀(κ))).exp(κ.cos(x-μ))

where I₀(κ) is a normalisation constant.

Using a uniform prior on μ over [0, 2.π)

and prior h₃(κ) = κ/(1+κ²)^3/2, then

Fisher information:

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) = d I₀(κ)/d κ

and A(κ) = d log(I₀(κ))/d κ = I₁(κ)/I₀(κ)