Binary Expansion Group Intersection Network
This provides a foundational method for graphical modeling beyond Gaussian assumptions, addressing a central challenge in statistics for multivariate data analysis.
The paper tackles the problem of characterizing conditional independence for non-Gaussian data by introducing the binary expansion group intersection network (BEGIN), a distribution-free graphical model for binary and bit-encoded multinomial variables, proving equivalences to sparse linear representations and matrix factorizations.
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.