A Parameterization-Invariant DIC

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.