A joint calibration model for combining predictive distributions

Abstract

In many research fields, as for example in probabilistic weather forecasting, valuable predictive information about a future random phenomenon may come from several, possibly heterogeneous, sources. Forecast combining methods have been developed over the years in order to deal with ensembles of sources: the aim is to combine several predictions in such a way to improve forecast accuracy and reduce risk of bad forecasts.
In this context, we propose the use of a Bayesian approach to information combining, which consists in treating the predictive probability density functions (pdfs) from the individual ensemble members as data in a Bayesian updating problem. The likelihood function is shown to be proportional to the product of the pdfs, adjusted by a joint “calibration function” describing the predicting skill of the sources (Morris, 1977). In this paper, after rephrasing Morris’ algorithm in a predictive context, we propose to model the calibration function in terms of bias, scale and correlation and to estimate its parameters according to the least squares criterion. The performance of our method is investigated and compared with that of Bayesian Model Averaging (Raftery, 2005) on simulated data.