Bayesian Nonparametric Calibration and Combination of Predictive Distributions

TL;DR

This paper presents a Bayesian nonparametric method for calibrating and combining predictive distributions, effectively handling model uncertainty and complex density shapes using Dirichlet process mixtures.

Contribution

It introduces a flexible Bayesian approach utilizing infinite beta mixtures for calibration, extending previous methods with nonparametric modeling and uncertainty quantification.

Findings

Successfully calibrates predictive densities with fat tails and multimodal features.

Demonstrates improved density forecast accuracy on financial and meteorological data.

Provides theoretical guarantees for posterior consistency.

Abstract

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.