Bayesian deviance ($D(\theta )=-2\log L(\theta )$)

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.