# Calibrated Multivariate Distributional Regression with Pre-Rank Regularization

arXiv: https://arxiv.org/abs/2601.22895 · HTML: https://mlsanity.com/paper/2601.22895/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 30 references that are in the map.
- First posted on arXiv 30 Jan 2026.

## Builds on (most cited first, 12 of 12)

- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Verified Uncertainty Calibration](https://mlsanity.com/paper/1909.10155/) (2019)
- [Localized Conformal Prediction: A Generalized Inference Framework for Conformal Prediction](https://mlsanity.com/paper/2106.08460/) (2021)
- [Soft Calibration Objectives for Neural Networks](https://mlsanity.com/paper/2108.00106/) (2021)
- [Probabilistic Conformal Prediction Using Conditional Random Samples](https://mlsanity.com/paper/2206.06584/) (2022)
- [Calibrated Model-Based Deep Reinforcement Learning](https://mlsanity.com/paper/1906.08312/) (2019)
- [A Large-Scale Study of Probabilistic Calibration in Neural Network Regression](https://mlsanity.com/paper/2306.02738/) (2023)
- [Score-based calibration testing for multivariate forecast distributions](https://mlsanity.com/paper/2211.16362/) (2022)
- [Assessing the calibration of multivariate probabilistic forecasts](https://mlsanity.com/paper/2307.05846/) (2023)
- [Enforcing tail calibration when training probabilistic forecast models](https://mlsanity.com/paper/2506.13687/) (2025)
- [Copula Calibration](https://mlsanity.com/paper/1307.7650/) (2013)
- [Multivariate Latent Recalibration for Conditional Normalizing Flows](https://mlsanity.com/paper/2505.16636/) (2025)

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

- [MEOM: Multi-View Expected-OKS Maximization for Human Pose Triangulation](https://mlsanity.com/paper/2608.30521/) (2026)
- [Towards Calibrated Probabilistic Forecasts for Events of Interest via Outcome-Conditional Recalibration](https://mlsanity.com/paper/2610.10076/) (2026)

## Abstract (arXiv)

The goal of probabilistic prediction is to issue predictive distributions that are as informative as possible, subject to being calibrated. Despite substantial progress in the univariate setting, achieving multivariate calibration remains challenging. Recent work has introduced pre-rank functions, scalar projections of multivariate forecasts and observations, as flexible diagnostics for assessing specific aspects of multivariate calibration, but their use has largely been limited to post-hoc evaluation. We propose a regularization-based calibration method that enforces multivariate calibration during training of multivariate distributional regression models using pre-rank functions. We further introduce a novel PCA-based pre-rank that projects predictions onto principal directions of the predictive distribution. Through simulation studies and experiments on 18 real-world multi-output regression datasets, we show that the proposed approach substantially improves multivariate pre-rank calibration without compromising predictive accuracy, and that the PCA pre-rank reveals dependence-structure misspecifications that are not detected by existing pre-ranks.