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

A note on the geometry of the MAP partition in Conjugate Exponential Bayesian Mixture Models