Bayesian Processor of Ensemble: A Gaussian-Gamma Model

            The Bayesian Processor of Ensemble (BPE) is a theoretically-based technique for probabilistic forecasting of weather variates.  It processes an ensemble of forecasts output from a numerical weather prediction (NWP) model and optimally fuses it with climatic data in order to quantify uncertainty about a predictand in the form of a posterior distribution function.  Using a family of such distribution functions, a given raw ensemble can be mapped into a posterior ensemble, which is well calibrated (against the climatic distribution of the predictand) at every point in space and time, has maximum informativeness, and preserves the spatio-temporal and inter-variate dependence structure of the NWP output fields.

            The Gaussian-Gamma model is the simplest version of the BPE.  It is suitable for (i) a predictand having a Gaussian prior (climatic) distribution, and (ii) an ensemble forecast having two sufficient statistics: (center, precision), where the precision, a measure of uncertainty, has a gamma distribution, and the center, a measure of central tendency, is stochastically dependent on the predictand and the precision, and has a Gaussian conditional distribution with a linear and separable dependence structure.  Surface temperature and its forecast from the Global Ensemble Forecast System (GEFS) of the National Centers for Environmental Prediction (NCEP) satisfy approximately these model assumptions.  Their samples (the climatic sample from 40 years, and the joint sample from 2 years) serve to illustrate (i) the estimation of model parameters, (ii) the validation of model assumptions, and (iii) the processing of an ensemble forecast through the BPE into the posterior density function, the posterior distribution function, the posterior quantile function, and the posterior ensemble — a coherent set of products which offers users complete information for making rational decisions via either analytic or simulation models.