Deviance Information Criterion for Bayesian model selection: Theoretical justification and applications

This paper provides a theoretical justification for the Deviance Information Criterion (DIC) as a Bayesian model selection tool using MCMC output. Unlike Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), which balance model adequacy against complexity without considering prior information, DIC incorporates priors into this trade-off. The contributions of this paper are two-fold. First, it demonstrates that when a plug-in predictive distribution — obtained by substituting parameter values with their optimal estimates to yield the plug-in estimated sampling distribution — is used under a set of regularity conditions, the DIC serves as an asymptotically unbiased estimator of the expected Kullback–Leibler divergence between the data-generating process and the plug-in predictive distribution. Second, it develops higher-order expansions for DIC and the effective number of parameters, highlighting the effect of the priors. We employ DIC to compare discrete-choice models, stochastic frontier models, and copula models in three empirical applications; the results align with theoretical expectations, showing the utility of DIC as a versatile tool outperforming the traditional model selection criteria. It is found that the logit model is better than the probit model for investigating the marginal effects of parents’ education on children’s completion of high school. Additionally, the stochastic frontier model with an exponential distribution better fits electricity utility data than the normal distribution. Finally, the chosen copula models for S&P index returns exhibit heavy tails and strong tail dependence. By modelling the effect of priors through higher order expansions, we also find the above empirical models outperforming their benchmark counterparts in terms of predictive accuracy.