Deviance Information Criterion

Definition

The Deviance Information Criterion (DIC) (Spiegelhalter, Best, Carlin, and van der Linde 2002) is a Bayesian model-comparison criterion for hierarchical models, designed to be computed trivially from MCMC output. Writing the deviance $D(\theta) = -2\log p(y\mid\theta)$, DIC combines a measure of fit (the posterior mean deviance) with a penalty for the effective number of parameters $p_D$: $$\mathrm{DIC} = \overline{D} + p_D = D(\bar\theta) + 2p_D,\qquad p_D = \overline{D} - D(\bar\theta),$$ where $\overline{D}=E_{\theta\mid y}[D(\theta)]$ is the posterior-mean deviance and $\bar\theta$ is the posterior mean of the parameters. Lower DIC indicates better expected out-of-sample predictive performance; like AIC it trades goodness-of-fit against complexity, but it estimates complexity from the posterior rather than by counting parameters.

Key Ideas

• Effective number of parameters. $p_D = \overline{D}-D(\bar\theta)$ measures how much the fit improves on average over the fit at the posterior mean — a data-driven "effective dimension" that automatically accounts for the shrinkage of hierarchical/random-effects parameters (so a model with many random effects counts far less than its nominal parameter count).
• Trivial to compute from MCMC. Both $\overline{D}$ (average $-2\log p(y\mid\theta^s)$ over draws) and $D(\bar\theta)$ come directly from posterior simulations — which is why DIC was built into BUGS/WinBUGS and became ubiquitous in applied Bayesian work.
• Relation to AIC. When the posterior is approximately normal and the prior is weak, $p_D\approx$ the number of parameters and DIC $\approx$ AIC; DIC generalizes AIC to hierarchical models where the parameter count is ambiguous.
• A plug-in, not fully Bayesian, criterion. DIC evaluates the fit term at the posterior point estimate $\bar\theta$, unlike WAIC, which averages the predictive density over the whole posterior.

Why It Matters

• The historical default. For a decade DIC was the standard automatic model-comparison number for Bayesian hierarchical models, because it required no extra computation beyond the MCMC already run.
• Superseded by WAIC / PSIS-LOO. More recent work (Vehtari–Gelman–Gabry 2017; Watanabe 2010) shows DIC's weaknesses and recommends WAIC or PSIS-LOO instead, so DIC is now mostly of historical and diagnostic interest.

Open Questions / Problems

• Not fully Bayesian. Because it plugs in the posterior mean $\bar\theta$, DIC is not invariant to reparametrization (a different parametrization gives a different $\bar\theta$ and hence a different DIC).
• Negative $p_D$. For some models the effective number of parameters comes out negative, which is uninterpretable and signals that the posterior-mean plug-in is a poor summary (e.g. strongly non-normal or multimodal posteriors, mixtures).
• Fails for singular models. DIC is problematic or undefined for singular models (mixtures, latent-variable models); Watanabe shows it is not asymptotically equivalent to Bayesian cross-validation there, unlike WAIC.
• Choice of focus. DIC depends on which parameters are treated as "in the model" (the level at which the likelihood is defined), so the same model can yield different DICs.

Related

• Widely Applicable Information Criterion
• Model Comparison
• Bayesian Model Selection
• Marginal Data Density
• Posterior Predictive Check
• Vehtari-Gelman-Gabry (2017)
• Watanabe (2010)