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.

翻译：条件独立性是现代统计学的核心概念，但在特殊参数族之外，它很少能精确表达为协方差特征。我们提出二进制展开群交网络（BEGIN），这是一种适用于多元二进制数据及比特编码多项变量的无分布图形表示方法。对于任意二进制随机向量及多项变量的比特表示，我们证明条件独立性等价于条件期望的稀疏线性表示、相应交互协方差矩阵的块分解以及关联广义舒尔补的块对角性。由此生成的图以二进制交互乘法群的交为索引，建立了高斯图形建模在高斯设定之外的模拟框架。该视角将数据比特视为原子，局部BEGIN分子视为大型马尔可夫随机场的构建模块。我们还展示了在温和正则条件下，二元比特表示如何使BEGIN能够逼近一般随机向量的条件独立性。关键技术手段是哈达玛棱镜——一种将交互协方差与群结构关联起来的线性映射。