Binary Cumulant Varieties

TL;DR

This paper explores the algebraic structure of binary cumulant varieties, demonstrating their advantages in representing complex statistical models like hyperdeterminants and secant varieties, and deriving related equations and inequalities.

Contribution

It introduces the use of binary cumulants for representing algebraic varieties in binary statistical models, providing new parametrizations and implicit equations for key varieties.

Findings

Derived parametrizations and equations for hyperdeterminants and secant varieties.

Showed advantages of cumulant representation in algebraic statistics.

Explored polynomial inequalities satisfied by cumulants.

Abstract

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.