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A Bayesian deviance is minus twice the log-likelihood, with any chosen additive data-only constant held consistent across the models being compared. Its Bayesian posterior expectation measures average fit. In the deviance information criterion, evaluating it at a parameter's posterior mean also enters the effective complexity penalty. This likelihood-based convention differs by a data-only constant from a saturated-model exponential-family deviance when a common saturated model exists.

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.

For the common Bayesian deviance convention $D=-2\log L$ used in all three models,Here Dhat is $D(\overline{\theta })$, an at-posterior-mean fit measure, and $p_D$ is an effective parameter count. The independent model's Dhat of 53.1 is almost identical to the exchangeable model's 53.2; both improve on the common model's 57.8. The common model's $p_D=7$ corresponds to six intercepts plus one shared effect. Independence uses roughly twelve effective parameters. Partial pooling reduces the exchangeable model's effective complexity to about 8.7 while retaining nearly the same fitted likelihood function as independence.

The exchangeable model has the lowest reported DIC, but the common model is competitive. Their difference is only about 1.3, whereas independence is worse by about 6.3. The deviance information criterion measures penalized fit for a predictive comparison, not model posterior probabilities, and these numbers do not establish overwhelming evidence for heterogeneity. The displayed exchangeable $\overline{D}+p_D$ is $61.9+8.7=70.6$, rather than the printed 70.5; rounding of the underlying values can account for a tenth and does not change this interpretation.