  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.
