# Binary Expansion Group Intersection Network

arXiv: https://arxiv.org/abs/2603.24763 · HTML: https://mlsanity.com/paper/2603.24763/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 15 references that are in the map.
- First posted on arXiv 25 Mar 2026.

## Builds on (most cited first, 7 of 7)

- [Structure Learning in Graphical Modeling](https://mlsanity.com/paper/1606.02359/) (2016)
- [BET on Independence](https://mlsanity.com/paper/1610.05246/) (2016)
- [Judicious Judgment Meets Unsettling Updating: Dilation, Sure Loss, and Simpson's Paradox](https://mlsanity.com/paper/1712.08946/) (2017)
- [Nonparametric Conditional Local Independence Testing](https://mlsanity.com/paper/2203.13559/) (2022)
- [Total positivity in exponential families with application to binary variables](https://mlsanity.com/paper/1905.00516/) (2019)
- [BEAUTY Powered BEAST](https://mlsanity.com/paper/2103.00674/) (2021)
- [BELIEF in Dependence: Leveraging Atomic Linearity in Data Bits for Rethinking Generalized Linear Models](https://mlsanity.com/paper/2210.10852/) (2022)

## Abstract (arXiv)

Conditional independence is central to modern statistics, but beyond special parametric families it rarely admits an exact covariance characterization. We introduce the binary expansion group intersection network (BEGIN), a distribution-free graphical representation for multivariate binary data and bit-encoded multinomial variables. For arbitrary binary random vectors and bit representations of multinomial variables, we prove that conditional independence is equivalent to a sparse linear representation of conditional expectations, to a block factorization of the corresponding interaction covariance matrix, and to block diagonality of an associated generalized Schur complement. The resulting graph is indexed by the intersection of multiplicative groups of binary interactions, yielding an analogue of Gaussian graphical modeling beyond the Gaussian setting. This viewpoint treats data bits as atoms and local BEGIN molecules as building blocks for large Markov random fields. We also show how dyadic bit representations allow BEGIN to approximate conditional independence for general random vectors under mild regularity conditions. A key technical device is the Hadamard prism, a linear map that links interaction covariances to group structure.