Bregman Divergencies, Triangle Inequality, and Maximum Likelihood Estimates for Normal Mixtures

The concepts of distance and angle and their algebraic realizations in the form of a scalar product are known to lead to kernel versions of sophisticated clustering algorithms. Here the more recently utilized Bregman Divergencies are treated. They possess all the properties of a metric apart from satisfying the triangle inequality. However, they can be suitably modified. En passant an apparent gap in a former paper is eliminated by exploiting an old proof concerning (conditionally) positive definite kernels. In addition an explicit isometric embedding of the modified Bregman Divergence in a Reproducing Kernel Hilbert Space is described. On a practical level recalling some basic facts on normal mixtures the well known connection between the parameter estimation problem for these mixtures and clustering algorithms is shown to hold in this abstract setting as well thus providing a more flexible approach to maximum likelihood estimates.