A note on the geometry of the MAP partition in some Normal Bayesian Mixture Models
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We investigate the geometry of the maximal a posteriori (MAP) partition in the Bayesian Mixture Model where the component distribution is multivariate Normal with Normal-inverse-Wishart prior on the component mean and covariance. We prove that in this case the clusters in any MAP partition are quadratically separable. Basically this means that every two clusters 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). We also describe a simple model where the covariance shape is fixed but there is a scaling parameter which may change from cluster to cluster. We prove that in any MAP partition for this model every two clusters are separated by an ellipsoid.
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