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20.1 Von Mises Distribution

20.1.1 Probability Density Function

If \(\mu \in \mathbb{R}\) and \(\kappa \in \mathbb{R}^+\), then for \(y \in \mathbb{R}\), \[ \text{VonMises}(y|\mu,\kappa) = \frac{\exp(\kappa\cos(y-\mu))}{2\pi I_0(\kappa)} \!. \] In order for this density to properly normalize, \(y\) must be restricted to some interval \((c, c + 2\pi)\) of length \(2 \pi\), because \[ \int_{c}^{c + 2\pi} \text{VonMises}(y|\mu,\kappa) dy = 1. \] Similarly, if \(\mu\) is a parameter, it will typically be restricted to the same range as \(y\).

If \(\kappa > 0\), a von Mises distribution with its \(2 \pi\) interval of support centered around its location \(\mu\) will have a single mode at \(\mu\); for example, restricting \(y\) to \((-\pi,\pi)\) and taking \(\mu = 0\) leads to a single local optimum at the mode \(\mu\). If the location \(\mu\) is not in the center of the support, the density is circularly translated and there will be a second local maximum at the boundary furthest from the mode. Ideally, the parameterization and support will be set up so that the bulk of the probability mass is in a continuous interval around the mean \(\mu\).

For \(\kappa = 0\), the Von Mises distribution corresponds to the circular uniform distribution with density \(1 / (2 \pi)\) (independently of the values of \(y\) or \(\mu\)).

20.1.2 Sampling Statement

y ~ von_mises(mu, kappa)

Increment target log probability density with von_mises_lpdf(y | mu, kappa) dropping constant additive terms.

20.1.3 Stan Functions

R von_mises_lpdf(reals y | reals mu, reals kappa)
The log of the von mises density of y given location mu and scale kappa. For a description of argument and return types, see section vectorized function signatures.

R von_mises_rng(reals mu, reals kappa)
Generate a Von Mises variate with location mu and scale kappa (i.e. returns values in the interval \([(\mu \mod 2\pi)-\pi,(\mu \mod 2\pi)+\pi]\)); may only be used in transformed data and generated quantities blocks. For a description of argument and return types, see section vectorized PRNG functions.

20.1.4 Numerical Stability

Evaluating the Von Mises distribution for \(\kappa > 100\) is numerically unstable in the current implementation. Nathanael I. Lichti suggested the following workaround on the Stan users group, based on the fact that as \(\kappa \rightarrow \infty\), \[ \text{VonMises}(y|\mu,\kappa) \rightarrow \text{Normal}(\mu, \sqrt{1 / \kappa}). \] The workaround is to replace y ~ von_mises(mu,kappa) with

 if (kappa < 100)
   y ~ von_mises(mu, kappa);
   y ~ normal(mu, sqrt(1 / kappa));