# Copula Calibration

arXiv: https://arxiv.org/abs/1307.7650 · HTML: https://mlsanity.com/paper/1307.7650/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jul 2013.

## Built on by (most cited first, 4 of 4)

- [Multivariate Latent Recalibration for Conditional Normalizing Flows](https://mlsanity.com/paper/2505.16636/) (2025)
- [Calibrated Multivariate Distributional Regression with Pre-Rank Regularization](https://mlsanity.com/paper/2601.22895/) (2026)
- [Enforcing Calibration in Multi-Output Probabilistic Regression with Pre-rank Regularization](https://mlsanity.com/paper/2510.21273/) (2025)
- [MEOM: Multi-View Expected-OKS Maximization for Human Pose Triangulation](https://mlsanity.com/paper/2608.30521/) (2026)

## Abstract (arXiv)

We propose notions of calibration for probabilistic forecasts of general multivariate quantities. Probabilistic copula calibration is a natural analogue of probabilistic calibration in the univariate setting. It can be assessed empirically by checking for the uniformity of the copula probability integral transform (CopPIT), which is invariant under coordinate permutations and coordinatewise strictly monotone transformations of the predictive distribution and the outcome. The CopPIT histogram can be interpreted as a generalization and variant of the multivariate rank histogram, which has been used to check the calibration of ensemble forecasts. Climatological copula calibration is an analogue of marginal calibration in the univariate setting. Methods and tools are illustrated in a simulation study and applied to compare raw numerical model and statistically postprocessed ensemble forecasts of bivariate wind vectors.