Binary Expansion Group Intersection Network

TL;DR

The paper introduces BEGIN, a novel distribution-free graphical model for multivariate binary data that characterizes conditional independence through linear and matrix factorizations, extending Gaussian graphical models.

Contribution

It develops a new group intersection network framework for binary data, linking conditional independence to algebraic and matrix properties, applicable beyond Gaussian assumptions.

Findings

BEGIN provides a sparse linear representation of conditional expectations.

The model characterizes conditional independence via block matrix factorizations.

Dyadic bit representations enable approximation of conditional independence in general data.

Abstract

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.