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Binary cumulant varieties. (English) Zbl 1274.13048

The authors study various algebraic varieties in the space of \(2\times 2\times \dots \times 2\)-tensors in terms of cumulant coordinates. If a tensor has non-negative real entries that sum up to \(1\), i.e., it is a probability distribution, then cumulants are quantities characterizing the distribution. In this article, cumulant coordinates are considered in the broader context of all \(2\times 2\times \dots \times 2\)-tensors, where they are given by the same change of coordinates as for probability distributions.
The authors show that the use of cumulant coordinates often simplifies the description of an algebraic variety in the space of \(2\times 2\times \dots \times 2\)-tensors. In particular, algebraic varieties defined by the hyperdeterminants, tangential and secant varieties of the Segre variety \((\mathbb{P}^1)^n\), hidden subset models and context-specific independence models are considered. The article concludes with the polynomial inequalities defining the semialgebraic set of cumulants coming from actual probability distributions.

MSC:

13P25 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)
05A40 Umbral calculus
14Q15 Computational aspects of higher-dimensional varieties
60C05 Combinatorial probability

Software:

Macaulay2
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References:

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