We impose rank one constraints on marginalizations of a tensor, given by a simplicial complex. Following work of Kirkup and Sullivant, such marginal independence models can be made toric by a linear change of coordinates. We study their toric ideals, with emphasis on random graph models and independent set polytopes of matroids. We develop the numerical algebra of parameter estimation, using both Euclidean distance and maximum likelihood, and we present a comprehensive database of small models.
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