  Mixed Membership Models (MMMs) are a popular family of latent structure
models for complex multivariate data. Instead of forcing each subject to belong
to a single cluster, MMMs incorporate a vector of subject-specific weights
characterizing partial membership across clusters. With this flexibility come
challenges in uniquely identifying, estimating, and interpreting the
parameters. In this article, we propose a new class of Dimension-Grouped MMMs
(Gro-M$^3$s) for multivariate categorical data, which improve parsimony and
interpretability. In Gro-M$^3$s, observed variables are partitioned into groups
such that the latent membership is constant for variables within a group but
can differ across groups. Traditional latent class models are obtained when all
variables are in one group, while traditional MMMs are obtained when each
variable is in its own group. The new model corresponds to a novel
decomposition of probability tensors. Theoretically, we derive transparent
identifiability conditions for both the unknown grouping structure and model
parameters in general settings. Methodologically, we propose a Bayesian
approach for Dirichlet Gro-M$^3$s to inferring the variable grouping structure
and estimating model parameters. Simulation results demonstrate good
computational performance and empirically confirm the identifiability results.
We illustrate the new methodology through applications to a functional
disability survey dataset and a personality test dataset.
