Identifiable Bayesian Deep Generative Copulas with Unknown Layer Widths for Data with Arbitrary Marginal Distributions

2026-05-26

AI Summary

This work addresses the lack of identifiability and interpretability in deep generative models when handling multivariate data with arbitrary marginal distributions. The authors propose an identifiable Bayesian deep generative Copula model based on a hierarchical binary latent variable structure. By employing a rank likelihood to decouple modeling of marginal distributions from dependence structure, the method achieves, for the first time, parameter identifiability in deep generative Copula models. A Bayesian rank selection prior with adaptive layer width is introduced to enhance flexibility. Theoretical analysis establishes posterior consistency under both continuous and mixed marginals, and the inference algorithm combines stochastic EM with maximum a posteriori estimation. Empirical results demonstrate superior performance in small-sample settings and reveal an interpretable hierarchical latent structure in personality survey data.

Abstract

Deep generative models offer powerful tools for multivariate data analysis, but their black-box architectures are often unidentified and difficult to interpret. We introduce the Deep Discrete Encoder (DDE) Copula, an identifiable and interpretable generative model for multivariate data with arbitrary marginal distributions. The model places a hierarchical directed network of binary latent variables inside a copula framework, enabling flexible dependence modeling for mixed discrete and continuous data. Estimation is based on rank likelihoods, which decouple marginal modeling from posterior inference on the DDE parameters and avoid specifying the marginal distributions. We establish conditions for identification of the DDE copula parameters, ensuring that layer-specific parameters provide meaningful summaries of multivariate dependence. We also prove quotient-space posterior consistency for continuous margins under the exact rank likelihood and treat the extended rank likelihood for tied or mixed margins as a generalized likelihood, with concentration under an additional contrast condition. For computation, we propose a stochastic expectation-maximization algorithm for \emph{maximum a posteriori} estimation, together with initialization strategies that improve convergence. To learn network dimension adaptively, we extend Bayesian rank-selection priors to infer layer-specific widths. Simulations show strong finite-sample performance, and a personality-survey analysis reveals interpretable hierarchical latent structure in complex multivariate data.

Problem

• identifiability
• deep generative models
• copula
• arbitrary marginal distributions
• interpretability

Innovation

• Identifiable generative model
• Deep Discrete Encoder Copula
• Rank likelihood
• Bayesian rank-selection priors
• Posterior consistency