  We propose a Bayesian test of normality for univariate or multivariate data
against alternative nonparametric models characterized by Dirichlet process
mixture distributions. The alternative models are based on the principles of
embedding and predictive matching. They can be interpreted to offer random
granulation of a normal distribution into a mixture of normals with mixture
components occupying a smaller volume the farther they are from the
distribution center. A scalar parametrization based on latent clustering is
used to cover an entire spectrum of separation between the normal distributions
and the alternative models. An efficient sequential importance sampler is
developed to calculate Bayes factors. Simulations indicate the proposed test
can detect non-normality without favoring the nonparametric alternative when
normality holds.
