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442 Environmental informaticsFIGURE 38.2Regions of unsustainable (> 2 ◦ Cwithpredictiveprobability.975)temperatureincrease obtained from pixel-wise predictive distributions, [Y (s)|Z], where s ∈North America.Since MCMC samples are taken from a (more than) 100,000-dimensionalposterior distribution, many such probability maps like Figure 38.2 can beproduced. For example, there is great interest in extreme temperature changes,so let k denote a temperature-change threshold; define the spatial probabilityfield,Pr(Y > k|Z), k ≥ 0. (38.21)As k increases in (38.21), the regions of North America that are particularlyvulnerable to climate change stand out. Decision-makers can query the BHMwhere, for NARCCAP, the query might involve the projected temperatureincrease in the 50 × 50 km pixel containing Columbus, Ohio. Or, it mightinvolve the projected temperature increase over the largely agricultural OlentangyRiver watershed (which contains Columbus). From (38.6), one can obtainthe probabilities (i.e., risks) of various projected climate-change scenarios,which represent real knowledge when weighing up mitigation and adaptationstrategies at the regional scale.This HM approach to uncertainty quantification opens up many possibilities:Notice that the occurrence-or-not of the events referred to above can bewritten as{∫ / ∫ }1{Y (s C ) >k} and 1 Y (s)ds ds >k ,OOwhere 1 is an indicator function, s C is the pixel containing Columbus, and O