Bayesian Statistics: An Introduction, 4th Edition

3.9 The circular normal distribution

3.9.1 Distributions on the circle

In this section, the variable is an angle running from to , that is, from 0 to radians. Such variables occur in a number of contexts, for example in connection with the homing ability of birds and in various problems in astronomy and crystallography. Useful references for such problems are Mardia (1972), Mardia and Jupp (2001), and Batschelet (1981). The method used here is a naïve numerical integration technique; for a modern approach using Monte Carlo Markov Chain (MCMC) methods, see Damian and Walker (1999).

The only distribution for such angles which will be considered is the so-called circular normal or von Mises’ distribution. An angle is said to have such a distribution with mean direction μ and concentration parameter κ if

and when this is so we write

The function is the modified Bessel function of the first ...