Bayesian Combination and Calibration of Predictive Distributions

Decision-makers often consult different experts to build a reliable forecast on some uncertain variable of interest. Combining more opinions and calibrating them to maximise the forecast accuracy is crucial issue treated also by Dawid (1982a), Dawid, DeGroot, Mortera (1995), Genest and Zidek (1986). A Bayesian approach was applied to predict a combined and calibrated density function using random calibration functionals and random combination weights. The linear, harmonic and logarithmic pools were used to explore the application of the Bayesian approach. Based on Gneiting and Ranjan (2013) and Bassetti, Casarin; Ravazzolo (2015) a beta mixture model was employed for the combination. The effects and techniques are demonstrated theoretically in simulation examples with multimodal densities.