Mixture Decomposition of Distributions using a Decomposition of the Sample Space

We consider the set of join probability distributions of $N$ binary random variables which can be written as a sum of $m$ distributions in the following form $p(x_1,\ldots,x_N)=\sum_{i=1}^m \alpha_i f_i(x_1,\ldots,x_N)$, where $\alpha_i \geq 0$, $\sum_{i=1}^m \alpha_i =1$, and the $f_i(x_1,\ldots,x_N)$ belong to some exponential family. For our analysis we decompose the sample space into portions on which the mixture components $f_i$ can be chosen arbitrarily. We derive lower bounds on the number of mixture components from a given exponential family necessary to represent distributions with arbitrary correlations up to a certain order or to represent any distribution. For instance, in the case where $f_i$ are independent distributions we show that every distribution $p$ on $\{0,1\}^N$ is contained in the mixture model whenever $m\geq 2^{N-1}$, and furthermore, that there are distributions which are not contained in the mixture model whenever $m<2^{N-1}$.