  We investigate the geometry of the maximal a posteriori (MAP) partition in
the Bayesian Mixture Model where the component and the base distributions are
chosen from conjugate exponential families. We prove that in this case the
clusters are separated by the contour lines of a linear functional of the
sufficient statistic. As a particular example, we describe Bayesian Mixture of
Normals with Normal-inverse-Wishart prior on the component mean and covariance,
in which the clusters in any MAP partition are separated by a quadratic
surface. In connection with results of Rajkowski (2018), where the linear
separability of clusters in the Bayesian Mixture Model with a fixed component
covariance matrix was proved, it gives a nice Bayesian analogue of the
geometric properties of Fisher Discriminant Analysis (LDA and QDA).
