# Mixtures and products in two graphical models

arXiv: https://arxiv.org/abs/1709.05276 · HTML: https://mlsanity.com/paper/1709.05276/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Sep 2017.

## Built on by (most cited first, 4 of 4)

- [Geometry of Linear Convolutional Networks](https://mlsanity.com/paper/2108.01538/) (2021)
- [Identifiability of Product of Experts Models](https://mlsanity.com/paper/2310.09397/) (2023)
- [Algebraic Invariants of Lightning Self-Attention](https://mlsanity.com/paper/2604.15632/) (2026)
- [Supermodular Rank: Set Function Decomposition and Optimization](https://mlsanity.com/paper/2305.14632/) (2023)

## Abstract (arXiv)

We compare two statistical models of three binary random variables. One is a mixture model and the other is a product of mixtures model called a restricted Boltzmann machine. Although the two models we study look different from their parametrizations, we show that they represent the same set of distributions on the interior of the probability simplex, and are equal up to closure. We give a semi-algebraic description of the model in terms of six binomial inequalities and obtain closed form expressions for the maximum likelihood estimates. We briefly discuss extensions to larger models.