Deviance information criterion ($\mathrm{DIC}=\overline{D}+p_D$)

For Bayesian deviance $D(\theta )=-2\log L(\theta )$ under a common likelihood convention, define $\overline{D}=\mathbb{E}[D(\theta )\mid y]$ and $p_D=\overline{D}-D(\mathbb{E}[\theta \mid y])$. Then $\mathrm{DIC}=\overline{D}+p_D$ trades average fit against effective complexity. Lower values suggest better penalized predictive fit in comparable models; differences are not Bayes factors or model posterior probabilities. The choice of stochastic parameters and latent-variable representation matters, especially in a hierarchical Bayesian model.

The DIC effective parameter count compares mean Bayesian deviance with deviance at the posterior mean. It need not be an integer or equal the number of coefficients. Interpretation depends on the parameterization and the chosen observational likelihood; nuisance intercepts count when they appear in that likelihood.

Suppose the Bayesian deviance is locally $D(\theta )=D(\hat{\theta })+(\theta -\hat{\theta }{)}^{T}J(\theta -\hat{\theta })$ with positive-definite $J$, and the posterior is normal with mean $m$ and covariance $\Sigma$. Writing $b=m-\hat{\theta }$, expectation and variance of the quadratic form giveConsequently the usual effective parameter count in DIC is $p_D=\mathrm{tr}(J\Sigma )$, whereas half the deviance variance is $\mathrm{tr}\{(J\Sigma {)}^2\}+2b^{T}J\Sigma Jb$. A locally flat prior gives $m=\hat{\theta }$ and $\Sigma =J^{-1}$, so both counts equal the parameter dimension. Informative priors generally destroy their equality. The expectation formula requires only the first two moments; normality is used for the variance formula.