A Parameterization-Invariant DIC

2026-05-26

AI Summary

This work addresses the limitations of the classical Deviance Information Criterion (DIC) in latent variable models, where reliance on a plug-in estimate based on the posterior mean renders it unstable—sometimes yielding negative effective parameter counts—and sensitive to parameterization due to non-identifiability. To overcome these issues, the authors propose a novel variant, DIC$_i$, which eliminates the need for plug-in estimation, is invariant to reparameterization, and avoids the pitfalls of the posterior mean in non-identifiable settings. Theoretical analysis demonstrates that DIC$_i$ is asymptotically equivalent to the Watanabe–Akaike Information Criterion (WAIC). Empirical evaluations in factor analysis and growth mixture models confirm that DIC$_i$ aligns closely with WAIC, whereas the classical DIC exhibits substantial inaccuracies, thereby validating the proposed criterion’s robustness and practical utility.

Abstract

The classic Deviance Information Criterion (DIC) is not invariant to reparameterization and can have a negative and unstable effective number of parameters. The reason for the effective number of parameters being negative is actually that the plug-in deviance becomes excessively large when the posterior means of the model parameter differ dramatically from the maximum likelihood estimates. In latent variable models, the cause can be identifiability issues that lead to meaningless and unstable plug-in estimates. Specifically, nonidentifiability means that distinct parameter points can have the same likelihood and switching between such points within or between MCMC chains produces unstable and meaningless posterior means. To address this issue, we propose a plug-in-free, parameterization-invariant version of the DIC, denoted DIC$_i$, and show that it is asymptotically equivalent to the Watanabe-Akaike Information Criterion (WAIC). Simulations demonstrate that DIC$_i$ aligns with WAIC in factor analysis and growth mixture models where the classic DIC breaks down. These results suggest that DIC$_i$ is a useful, computationally efficient alternative to the DIC when WAIC is not applicable or not available.

Problem

• Deviance Information Criterion
• parameterization invariance
• effective number of parameters
• nonidentifiability
• latent variable models

Innovation

• parameterization-invariant
• DIC
• WAIC
• latent variable models
• nonidentifiability