  For predictive evaluation based on quasi-posterior distributions, we develop
a new information criterion, the posterior covariance information criterion
(PCIC. PCIC generalises the widely applicable information criterion WAIC so as
to effectively handle predictive scenarios where likelihoods for the estimation
and the evaluation of the model may be different. A typical example of such
scenarios is the weighted likelihood inference, including prediction under
covariate shift and counterfactual prediction. The proposed criterion utilises
a posterior covariance form and is computed by using only one Markov chain
Monte Carlo run. Through numerical examples, we demonstrate how PCIC can apply
in practice. Further, we show that PCIC is asymptotically unbiased to the
quasi-Bayesian generalization error under mild conditions in weighted inference
with both regular and singular statistical models.
