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.

Missing values with mixed data types is a common problem in a large number of machine learning applications such as processing of surveys and in different medical applications. Recently, Gaussian copula models have been suggested as a means of performing imputation of missing values using a probabilistic framework. While the present Gaussian copula models have shown to yield state of the art performance, they have two limitations: they are based on an approximation that is fast but may be imprecise and they do not support unordered multinomial variables. We address the first limitation using direct and arbitrarily precise approximations both for model estimation and imputation by using randomized quasi-Monte Carlo procedures. The method we provide has lower errors for the estimated model parameters and the imputed values, compared to previously proposed methods. We also extend the previous Gaussian copula models to include unordered multinomial variables in addition to the present support of ordinal, binary, and continuous variables.