Geodesic
Mixture decompositions of exponential families using a decomposition of their sample spaces
Kybernetika, Tome 49 (2013) no. 1, pp. 23-39 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We study the problem of finding the smallest $m$ such that every element of an exponential family can be written as a mixture of $m$ elements of another exponential family. We propose an approach based on coverings and packings of the face lattice of the corresponding convex support polytopes and results from coding theory. We show that $m=q^{N-1}$ is the smallest number for which any distribution of $N$ $q$-ary variables can be written as mixture of $m$ independent $q$-ary variables. Furthermore, we show that any distribution of $N$ binary variables is a mixture of $m = 2^{N-(k+1)}(1+ 1/(2^k-1))$ elements of the $k$-interaction exponential family.
We study the problem of finding the smallest $m$ such that every element of an exponential family can be written as a mixture of $m$ elements of another exponential family. We propose an approach based on coverings and packings of the face lattice of the corresponding convex support polytopes and results from coding theory. We show that $m=q^{N-1}$ is the smallest number for which any distribution of $N$ $q$-ary variables can be written as mixture of $m$ independent $q$-ary variables. Furthermore, we show that any distribution of $N$ binary variables is a mixture of $m = 2^{N-(k+1)}(1+ 1/(2^k-1))$ elements of the $k$-interaction exponential family.
Classification : 52B05, 60C05, 62E17
Keywords: mixture model; non-negative tensor rank; perfect code; marginal polytope 
@article{KYB_2013_49_1_a2,
     author = {Mont\'ufar, Guido},
     title = {Mixture decompositions of exponential families using a decomposition of their sample spaces},
     journal = {Kybernetika},
     pages = {23--39},
     year = {2013},
     volume = {49},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2013_49_1_a2/}
}
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Montúfar, Guido. Mixture decompositions of exponential families using a decomposition of their sample spaces. Kybernetika, Tome 49 (2013) no. 1, pp. 23-39. http://geodesic.mathdoc.fr/item/KYB_2013_49_1_a2/