# Bayesian Nonparametric Calibration and Combination of Predictive Distributions

arXiv: https://arxiv.org/abs/1502.07246 · HTML: https://mlsanity.com/paper/1502.07246/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Feb 2015.

## Built on by (most cited first, 6 of 6)

- [Forecast combinations: an over 50-year review](https://mlsanity.com/paper/2205.04216/) (2022)
- [A review of predictive uncertainty estimation with machine learning](https://mlsanity.com/paper/2209.08307/) (2022)
- [Bayesian Nonlinear Regression using Sums of Simple Functions](https://mlsanity.com/paper/2312.01881/) (2023)
- [Predictive Density Combination Using a Tree-Based Synthesis Function](https://mlsanity.com/paper/2311.12671/) (2023)
- [AdaPRL: Adaptive Pairwise Regression Learning with Uncertainty Estimation for Universal Regression Tasks](https://mlsanity.com/paper/2501.05809/) (2025)
- [Statistical versus machine learning-based spatial interpolation of post-processed ensemble weather forecasts](https://mlsanity.com/paper/2609.07512/) (2026)

## Abstract (arXiv)

We introduce a Bayesian approach to predictive density calibration and combination that accounts for parameter uncertainty and model set incompleteness through the use of random calibration functionals and random combination weights. Building on the work of Ranjan, R. and Gneiting, T. (2010) and Gneiting, T. and Ranjan, R. (2013), we use infinite beta mixtures for the calibration. The proposed Bayesian nonparametric approach takes advantage of the flexibility of Dirichlet process mixtures to achieve any continuous deformation of linearly combined predictive distributions. The inference procedure is based on Gibbs sampling and allows accounting for uncertainty in the number of mixture components, mixture weights, and calibration parameters. The weak posterior consistency of the Bayesian nonparametric calibration is provided under suitable conditions for unknown true density. We study the methodology in simulation examples with fat tails and multimodal densities and apply it to density forecasts of daily S&P returns and daily maximum wind speed at the Frankfurt airport.