Posterior Covariance Information Criterion for Weighted Inference
This work addresses predictive evaluation challenges in quasi-Bayesian inference, particularly for weighted scenarios like covariate shift, but it is incremental as it builds on existing criteria like WAIC.
The authors tackled the problem of predictive evaluation for quasi-posterior distributions by developing the posterior covariance information criterion (PCIC), which generalizes WAIC to handle scenarios with different likelihoods for estimation and evaluation, such as weighted likelihood inference. They demonstrated its practical applicability through numerical examples and proved it is asymptotically unbiased under mild conditions.
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.