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Dimension-Grouped Mixed Membership Models for Multivariate Categorical Data.

Yuqi Gu1, Elena A Erosheva2, Gongjun Xu3

  • 1Department of Statistics Columbia University New York, NY 10027, USA.

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|April 17, 2025
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Summary
This summary is machine-generated.

Dimension-Grouped Mixed Membership Models (DGMMMs) offer improved parsimony and interpretability for complex multivariate categorical data. This new approach enhances parameter identification and estimation for latent structure analysis.

Keywords:
Bayesian MethodsGrade of Membership ModelIdentifiabilityMixed Membership ModelMultivariate Categorical DataProbabilistic Tensor Decomposition

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Area of Science:

  • Multivariate Statistics
  • Latent Variable Modeling
  • Categorical Data Analysis

Background:

  • Mixed Membership Models (MMMs) provide flexible partial subject clustering for complex data.
  • Traditional MMMs face challenges in parameter identification, estimation, and interpretation.
  • Existing latent class models assume single cluster membership, limiting flexibility.

Purpose of the Study:

  • Introduce Dimension-Grouped Mixed Membership Models (DGMMMs) for multivariate categorical data.
  • Enhance parsimony and interpretability compared to traditional MMMs.
  • Address challenges in identifying, estimating, and interpreting MMM parameters.

Main Methods:

  • Proposed DGMMMs partition observed variables into groups with constant latent membership within groups.
  • Developed a novel probability tensor decomposition for the DGMM framework.
  • Derived theoretical identifiability conditions for grouping structure and model parameters.
  • Implemented a Bayesian approach using Dirichlet priors for inference.

Main Results:

  • Theoretical derivations provide transparent identifiability conditions.
  • Simulation studies show good computational performance and confirm identifiability.
  • The DGMM framework offers a novel decomposition of probability tensors.
  • Bayesian inference effectively estimates model parameters and infers grouping structure.

Conclusions:

  • DGMMMs offer a more parsimonious and interpretable alternative for multivariate categorical data analysis.
  • The proposed methodology successfully addresses identifiability and estimation challenges in MMMs.
  • Demonstrated practical utility through applications in disability and personality datasets.