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We study the problem of finding the smallest $m$ such that every element ofan exponential family can be written as a mixture of $m$ elements of anotherexponential family. We propose an approach based on coverings and packings ofthe face lattice of the corresponding convex support polytopes and results fromcoding theory. We show that $m=q^{N-1}$ is the smallest number for which anydistribution 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 $m = 2^{N-(k+1)}(1+ 1/(2^k-1))$ elementsof the $k$-interaction exponential family.

Mixture decompositions of exponential families using a decomposition of their sample spaces