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
DOI: 10.1145/3597066.3597084 ARXIV: https://arxiv.org/abs/2301.09068 CODE: https://mathrepo.mis.mpg.de/MomentVarietiesForMixturesOfProducts

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.

Summary

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.

Code

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, yulia@math.berkeley.edu