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.
Highlights
Combining forecasts from different statistical models or other sources of information is a crucial problem in many important applications
We propose a flexible Bayesian nonparametric approach to calibration and combination that relies on beta mixtures, and nests the beta transformed linear pool introduced by Ranjan and Gneiting (2010) and Gneiting and Ranjan (2013)
We note that the uncertainty of the number components in the infinite beta mixture implies a wider high probability density region (HPD), see gray lines in 1, than that given by the finite beta mixture calibration, see third panel in 8
Summary
Combining forecasts from different statistical models or other sources of information is a crucial problem in many important applications. We provide some conditions under which the proposed probabilistic calibration converges in terms of weak posterior consistency to the true underlying density as the number of observations goes to infinity This calibration property is a very powerful result, which substantially improves upon the earlier approach of Gneiting and Ranjan (2013), which was shown to be flexibly dispersive only in the sense of second moment of the probabilistic forecast.
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