Dimension-Grouped Mixed Membership Models for Multivariate Categorical Data
Authors: Yuqi Gu, Elena E. Erosheva, Gongjun Xu, David B. Dunson; Volume 24, Issue 88, Pages 1-49, 2023.
Abstract
Mixed Membership Models (MMMs) are widely used latent structure models for complex multivariate data. Instead of assigning each subject to a single cluster, MMMs incorporate subject-specific weights that represent partial membership across clusters. While this flexibility is advantageous, it also presents challenges in terms of parameter identification, estimation, and interpretation. In this article, we introduce a new class of Dimension-Grouped MMMs (Gro-M$^3$s) for analyzing multivariate categorical data. These models improve parsimony and interpretability by partitioning observed variables into groups based on shared latent membership. Within each group, the latent membership is constant, but it can differ across groups. Traditional latent class models are obtained when all variables are in the same group, while traditional MMMs are obtained when each variable is in its own group. The Gro-M$^3$ model corresponds to a novel decomposition of probability tensors. We derive transparent identifiability conditions for both the unknown grouping structure and model parameters in general settings. We also propose a Bayesian approach for inferring the variable grouping structure and estimating model parameters using Dirichlet Gro-M$^3$s. 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.
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