Local Mixture Model in Hilbert Space
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In this thesis, we study local mixture models with a Hilbert space structure. First, we consider the fibre bundle structure of local mixture models in a Hilbert space. Next, the spectral decomposition is introduced in order to construct local mixture models. We analyze the approximation error asymptotically in the Hilbert space. After that, we will discuss the convexity structure of local mixture models. There are two forms of convexity conditions to consider, first due to positivity in the $-1$-affine structure and the second by points having to lie inside the convex hull of a parametric family. It is shown that the set of mixture densities is located inside the intersection of the sets defined by these two convexities. Finally, we discuss the impact of the approximation error in the Hilbert space when the domain of mixing variable changes.