Moment Varieties for Mixtures of Products

This page contains auxiliary files to the paper:
Yulia Alexandr, Joe Kileel, and Bernd Sturmfels: Moment varieties for mixtures of products
In: ISSAC ‘23 proceedings of the international symposium on symbolic and algebraic computation ; Tromso Norway ; July 24-27, 2023 / Alicia Dickenstein … (eds.)
New York : ACM, 2023 . - P. 53-60

ABSTRACT: The setting of this article is nonparametric algebraic statistics. We study moment varieties of conditionally independent mixture distributions on \(\mathbb{R}^n\). These are the secant varieties of toric varieties that express independence in terms of univariate moments. Our results revolve around the dimensions and defining polynomials of these varieties.


Consider \(n\) independent random variables \(X_1,X_2,\ldots,X_n\) on \(\mathbb{R}\). We make no assumptions about the \(X_k\) other than that their moments \(\mu_{ki} = \mathbb{E}(X_k^i)\) exist. We write \(m_{i_1i_2\ldots i_n}\) for the moments of the random vector \(X=(X_1,X_2\ldots,X_n)\). By independence, we have

\[\begin{align*} m_{i_1 i_2 \cdots i_n}= \mathbb{E}(X_1^{i_1} X_2^{i_2} \cdots X_n^{i_n}) = \mathbb{E}(X_1^{i_1} ) \mathbb{E}(X_2^{i_2}) \cdots \mathbb{E}(X_n^{i_n}) = \mu_{1 i_1} \mu_{2 i_2} \cdots \mu_{n i_n} . \end{align*}\]

We consider the squarefree parametrization for the moments with \(i_1 + i_2 + \cdots + i_n = d\), whose image is the toric variety \(\mathcal{M}_{n,d}\). Furthermore, we study mixtures of \(r\) independent distributions and their images under certain coordinate projections. The associated moment varieties are the secant varieties \(\sigma_r(\mathcal{M}_{n,d})\) and \(\sigma_r(\mathcal{M}_{n,\lambda})\), respectively.


Our code is written in \(\verb|Macaulay2|\), \(\verb|Julia|\), and \(\verb|Maple|\). We use \(\verb|Macaulay2|\) to compute dimensions of the secant varieties we study. We also compute the degrees of several secant varieties, both numerically and symbolically.

We also include the code used to prove Proposition 34, implemented in \(\verb|Maple|\), the code to form a masked Hankel flattening matrix, the code to compute toric ideals of the varieties when \(r=1\), and the code relevant for the finiteness results in Section 3.

Project page created: 10/05/2023.
Project contributors: Yulia Alexandr, Joe Kileel, and Bernd Sturmfels.

Software used: Macaulay2 (version 1.17), Julia (version 1.8), Maple.
System setup used: MacBook Pro with macOS BigSur 11.1, Chip Apple M1, Memory 8 GB.

Corresponding author of this page: Yulia Alexandr,