Journal of Data Science 11(2013), 819-833 A Bayesian Analysis of the Spherical Distribution in Presence of Covariates Jorge Alberto Achcar 1 , Gian Franco Napa 1 and Roberto Molina De Souza 2* 1 University of S˜ao Paulo and 2 Federal Technological University of Paran´a Abstract: In this paper we introduce a Bayesian analysis of a spherical distri- bution applied to rock joint orientation data in presence or not of a vector of covariates, where the response variable is given by the angle from the mean and the covariates are the components of the normal upwards vector. Standard simulation MCMC (Markov Chain Monte Carlo) methods have been used to obtain the posterior summaries of interest obtained from Win- Bugs software. Illustration of the proposed methodology are given using a simulated data set and a real rock spherical data set from a hydroelectrical site. Key words: Bayesian analysis, MCMC methods, regression models, rock spherical engineering data, spherical distribution. 1. Introduction A distribution of errors over the surface of the unit sphere was introduced by R. A. Fisher in 1952, given by the density, f (θ | k)= k 2 sinh (k) exp(k cos (θ)) sin (θ) , (1) where 0 <θ<π. The random variable θ is the angular displacement from the true position at which θ = 0; the parameter k is a measure of precision. The parameter k has the following interpretation: (i) If k is large the distribution is confined in a small portion of the sphere in the neighborhood of the maximum (small variability); (ii) If k = 0 (great variability), the distribution is uniform over all spherical surface. * Corresponding author.