Algebraic statistics for binary random variables is concerned with highly structured algebraic varieties in the space of 2x2x...x2-tensors. We demonstrate the advantages of representing such varieties in the coordinate system of binary cumulants. Our primary focus lies on hidden subset models. Parametrizations and implicit equations in cumulants are derived for hyperdeterminants, for secant and tangential varieties of Segre varieties, and for certain context-specific independence models. Extending work of Rota and collaborators, we explore the polynomial inequalities satisfied by cumulants.
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