  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.
