A Bayesian interpretation of the multivariate skew-normal distribution

This paper provides a unified treatment and a Bayesian interpretation of two different classes of multivariate skew-normal distributions proposed by Azzalini and Dalla Valle (Biometrika 83 (1996) 715) and Gupta et al. (Tech. Rep., Cimat, Mexico (2001)). We show that the above classes of distributions can be viewed as particular cases of a more general family, which naturally arise in constrained modelling. Our approach can be viewed as a direct extension to the multivariate case of the O'Hagan and Leonard (Biometrika 63 (1976) 201) paper, where the authors construct, in the scalar case, a skew prior distribution for the location parameter of a Gaussian random variable, using a simple hierarchical argument.