# Moment Varieties for Mixtures of Products

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

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

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, yulia@math.berkeley.edu