Deviance information criterion

The deviance information criterion (DIC) is a hierarchical modeling generalization of the Akaike information criterion (AIC). It is particularly useful in Bayesian model selection problems where the posterior distributions of the models have been obtained by Markov chain Monte Carlo (MCMC) simulation. DIC is an asymptotic approximation as the sample size becomes large, like AIC. It is only valid when the posterior distribution is approximately multivariate normal. Define the deviance as , where are the data, are the unknown parameters of the model and is the likelihood function. is a constant that cancels out in all calculations that compare different models, and which therefore does not need to be known. There are two calculations in common usage for the effective number of parameters of the model. The first, as described in , is , where is the expectation of . The second, as described in , is . The larger the effective number of parameters is, the easier it is for the model to fit the data, and so the deviance needs to be penalized. The deviance information criterion is calculated as or equivalently as From this latter form, the connection with AIC is more evident. The idea is that models with smaller DIC should be preferred to models with larger DIC. Models are penalized both by the value of , which favors a good fit, but also (similar to AIC) by the effective number of parameters . Since will decrease as the number of parameters in a model increases, the term compensates for this effect by favoring models with a smaller number of parameters. An advantage of DIC over other criteria in the case of Bayesian model selection is that the DIC is easily calculated from the samples generated by a Markov chain Monte Carlo simulation. AIC requires calculating the likelihood at its maximum over , which is not readily available from the MCMC simulation. But to calculate DIC, simply compute as the average of over the samples of , and as the value of evaluated at the average of the samples of . Then the DIC follows directly from these approximations.