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MiS Preprint

Mixture Decomposition of Distributions using a Decomposition of the Sample Space

Guido Montúfar


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 $m<2^{N-1}$.

Aug 1, 2010
Aug 4, 2010

Related publications

2013 Journal Open Access
Guido Montúfar

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

In: Kybernetika, 49 (2013) 1, pp. 23-39