Integrated Deviance Information Criterion for Latent Variable Models

Deviance information criterion (DIC) has been widely used for Bayesian model comparison, especially after Markov chain Monte Carlo (MCMC) is used to estimate candidate models. This paper studies the problem of using DIC to compare latent variable models after the models are estimated by MCMC together with the data augmentation technique. Our contributions are twofold. First, we show that when MCMC is used with data augmentation, it undermines theoretical underpinnings of DIC. As a result, by treating latent variables as parameters, the widely used way of constructing DIC based on the conditional likelihood, although facilitating computation, should not be used. Second, we propose two versions of integrated DIC (IDIC) to compare latent variable models without treating latent variables as parameters. The large sample properties of IDIC are studied and an asymptotic justi fication of IDIC is provided. Some popular algorithms such as the EM, Kalman and particle filtering algorithms are introduced to compute IDIC for latent variable models. IDIC is illustrated using asset pricing models, dynamic factor models, and stochastic volatility models.