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，它很少能精确表征为协方差形式。我们提出二元展开群交网络（BEGIN），这是一种适用于多变量二元数据和比特编码多项变量的无分布图形表示方法。对于任意二元随机向量和多项变量的比特表示，我们证明了条件独立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。由此产生的图由二元交互乘法群的交集索引，从而在高斯框架之外构建了高斯图模型的类比。该观点将数据比特视为原子，将局部BEGIN分子视为大型马尔可夫随机场的构建基块。我们还展示了在温和正则性条件下，二元比特表示如何使BEGIN能够近似一般随机向量的条件独立。核心技术手段是哈达玛棱镜，这是一种将交互协方差与群结构相连接的线性映射。