Trigonometric moment matching and minimization of the Kullback-Leibler divergence

We show an important property of the von Mises distribution on the unit circle. If we approximate an arbitrary circular distribution using a von Mises distribution, the result obtained by trigonometric moment matching also minimizes the Kullback-Leibler divergence (Theorem 1). This result is a justification for circular filtering algorithms based on trigonometric moment matching as the loss of information is minimized. Furthermore, we show that Theorem 1 does not hold for the wrapped normal distribution.