  With the advent of ubiquitous monitoring and measurement protocols, studies
have started to focus more and more on complex, multivariate and heterogeneous
datasets. In such studies, multivariate response variables are drawn from a
heterogeneous population often in the presence of additional covariate
information. In order to deal with this intrinsic heterogeneity, regression
analyses have to be clustered for different groups of units. Up until now,
mixture model approaches assigned units to distinct and non-overlapping groups.
However, not rarely these units exhibit more complex organization and
clustering. It is our aim to define a mixture of generalized linear models with
overlapping clusters of units. This involves crucially an overlap function,
that maps the coefficients of the parent clusters into the the coefficient of
the multiple allocation units. We present a computationally efficient MCMC
scheme that samples the posterior distribution of the parameters in the model.
An example on a two-mode network study shows details of the implementation in
the case of a multivariate probit regression setting. A simulation study shows
the overall performance of the method, whereas an illustration of the voting
behaviour on the US supreme court shows how the 9 justices split in two
overlapping sets of justices.
