  In this paper, a novel Bayesian nonparametric test for assessing multivariate
normal models is presented. While there are extensive frequentist and graphical
methods for testing multivariate normality, it is challenging to find Bayesian
counterparts. The proposed approach is based on the use of the Dirichlet
process and Mahalanobis distance. More precisely, the Mahalanobis distance is
employed as a good technique to transform the $m$-variate problem into a
univariate problem. Then the Dirichlet process is used as a prior on the
distribution of the Mahalanobis distance. The concentration of the distribution
of the distance between the posterior process and the chi-square distribution
with $m$ degrees of freedom is compared to the concentration of the
distribution of the distance between the prior process and the chi-square
distribution with $m$ degrees of freedom via a relative belief ratio. The
distance between the Dirichlet process and the chi-square distribution is
established based on the Anderson-Darling distance. Key theoretical results of
the approach are derived. The procedure is illustrated through several
examples, in which the proposed approach shows excellent performance.
