A Family of Blockwise One-Factor Distributions for Modelling High-Dimensional Binary Data

We introduce a new family of one factor distributions for high-dimensional binary data. The model provides an explicit probability for each event, thus avoiding the numeric approximations often made by existing methods. Model interpretation is easy since each variable is described by two continuous parameters (corresponding to its marginal probability and to its strength of dependency with the other variables) and by one binary parameter (defining if the dependencies are positive or negative). An extension of this new model is proposed by assuming that the variables are split into independent blocks which follow the new one factor distribution. Parameter estimation is performed by the inference margin procedure where the second step is achieved by an expectation-maximization algorithm. Model selection is carried out by a deterministic approach which strongly reduces the number of competing models. This approach uses a hierarchical ascendant classification of the variables based on the empirical version of Cramer's V for selecting a narrow subset of models. The consistency of such procedure is shown. The new model is evaluated on numerical experiments and on a real data set. The procedure is implemented in the R package MvBinary available on CRAN.