Aust. N. Z. J. Stat. 42(4), 2000, 433–440 AN EMPIRICAL BAYES INFERENCE FOR THE VON MISES DISTRIBUTION JOSEMAR RODRIGUES 1∗ ,JOS ´ E GALV ˜ AO LEITE 1 AND LUIS A. MILAN 1 DEs-UFSCar-S˜ ao Carlos-Brazil Summary This paper develops an empirical Bayesian analysis for the von Mises distribution, which is the most useful distribution for statistical inference of angular data. A two-stage infor- mative prior is proposed, in which the hyperparameter is obtained from the data in one of the stages. This empirical or approximate Bayes inference is justified on the basis of maximum entropy, and it eliminates the modified Bessel functions. An example with real data and a realistic prior distribution for the regression coefficients is considered via a Metropolis-within-Gibbs algorithm. Key words: angular data; link function; maximum entropy; Metropolis-within-Gibbs algorithm; regression models. 1. Introduction Bagchi & Guttman (1988) considered a Bayesian analysis of the multivariate von Mises distribution and developed theorems that are analogous to the theorems of Lindley & Smith (1972). They formulated a conjugated prior for the mean direction and a non-informative prior for the concentration parameter which maximize the entropy, subject to constraints on the first moments and assuming that the density integrates to one (see Bagchi, 1987 p . 23). In this article, we consider a different Bayesian approach to the univariate von Mises distribution in the sense that the prior distributions are chosen so that the entropy of the posterior distribu- tions is maximized. The von Mises distribution is the most useful distribution for statistical inference of angular data and has many of the key properties for statistical inference that the normal distribution has for linear data. A circular random variable θ is said to have a von Mises distribution vM(µ,λ) if its probability density function is given by f(θ | µ,λ) = 1 2πI 0 (λ) exp ( λ cos(θ - µ) ) (0 ≤ θ ≤ 2π), (1) where µ is the mean direction (0 ≤ µ ≤ 2π),λ is the concentration parameter (λ ≥ 0) and I 0 (λ) denotes the modified Bessel function of the first kind and order zero (see e.g. Fisher, 1993). The value λ = 0 corresponds to the circular uniform distribution; for λ> 0, the density is unimodal and symmetrical about µ, and increasingly concentrated as λ increases. In Section 2, motivated by Johnson & Wehrly (1978), we suggest a two-stage prior for µ and λ for which the joint posterior distribution is the maximum entropy distribution, subject Received September 1998; revised December 1999; accepted December 1999. ∗ Author to whom correspondence should be addressed. 1 DEs-UFSCar-CP 676-S˜ ao Carlos-SP-CEP 13565-905, Brazil. e-mail: vjosemar@power.ufscar.br Acknowledgments. The authors thank an Associate Editor and the referee for suggestions and comments which improved this paper. This work was supported by Conselho Nacional de Desenvolvimento Cient´ ıfico e Tecnol´ ogico (CNPq), Brazil. c  Australian Statistical Publishing Association Inc. 2000. Published by Blackwell Publishers Ltd, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden MA 02148, USA