# Posterior Covariance Information Criterion for Weighted Inference

arXiv: https://arxiv.org/abs/2106.13694 · HTML: https://mlsanity.com/paper/2106.13694/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 31 references that are in the map.
- First posted on arXiv 25 Jun 2021.

## Builds on (most cited first, 2 of 2)

- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC](https://mlsanity.com/paper/1507.04544/) (2015)

## Built on by (most cited first, 2 of 2)

- [Review and Prospect of Algebraic Research in Equivalent Framework between Statistical Mechanics and Machine Learning Theory](https://mlsanity.com/paper/2406.10234/) (2024)
- [Variational Gaussian Approximation in Replica Analysis of Parametric Models](https://mlsanity.com/paper/2509.11780/) (2025)

## Abstract (arXiv)

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.