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Binary Cumulant Varieties

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Abstract

Algebraic statistics for binary random variables is concerned with highly structured algebraic varieties in the space of 2×2×···×2-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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Correspondence to Piotr Zwiernik.

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This work was conducted during the Spring 2011 program “Algebraic Geometry with a View Towards Applications" at the Institute Mittag-Leffler in Djursholm, Sweden. The first author was partially supported by KTH Stockholm and the US National Science Foundation (DMS-0968882). The second author gratefully acknowledges support from the AXA Mittag-Leffler Fellowship Project, sponsored by the AXA Research Fund.

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Sturmfels, B., Zwiernik, P. Binary Cumulant Varieties. Ann. Comb. 17, 229–250 (2013). https://doi.org/10.1007/s00026-012-0174-1

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  • DOI: https://doi.org/10.1007/s00026-012-0174-1

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