Comparative Forecast Evaluation: Graphical Gaussian Models and Sufficiency Relations

The sufficiency analysis of DeGroot and Fienberg (1983) compares the structures of the bivariate probability distributions p(u, fa) and p(u, fb) and does not investigate the full trivariate distribution p(u, fa, f b) like the graphical modeling approach does. [...]forecasts f a being sufficient for f b does not preclude that more information can be gained from learning the predictions f b in addition to the predictions f a. According to the general theory the forecasts f a are sufficient for the forecasts f b if a stochastic transformation between bivariate distributions can be established, which indicates an increase of uncertainty. Knowing historical records of the forecasts f delivered for scalar events u, any rational would employ the conditional (or posterior) expectation, E(u | f ), as a calibration filter to compensate for systematic forecast errors (Murphy et al. 1989; Krzysztofowicz 1992). Since the posterior distribution p(u | f ) is normal, the filter is linear in u: E(u | f ) 5 Bu,f f with Bu,f 5 cov(u, f) var21( f ), (1) with Bu,f denoting the least squares regression coefficient.