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432 Environmental informaticsthis with respect to Ŷ . Then it is a consequence of decision theory (Berger,1985) that the optimal decision isY ∗ (Z) = arg inf E{L(Y,Ŷ )|Z}, (38.7)Ŷwhere for some generic function g,thenotationE{g(Y )|Z} is used to representthe conditional expectation of g(Y ) given Z.Sometimes E{L(Y,Ŷ )|θ} is called the risk, but I shall call it the expectedloss; sometimes E{L(Y,Ŷ )} is called the Bayes risk, but see above where I havecalled it the Bayes expected loss. In what follows, I shall reserve the word riskto be synonymous with probability.Now, if θ were known, only Y remains unknown, and HM involves just(38.1)–(38.2). Then Bayes’ Theorem yields[Y |Z, θ] =[Z|Y,θ] × [Y |θ]/[Z|θ]. (38.8)In this circumstance, (38.8) is both the posterior distribution and the predictivedistribution; because of the special role of Y ,Iprefertocallitthepredictive distribution. The analogue to (38.7) when θ is known is, straightforwardly,Y ∗ (Z) = arg inf E{L(Y,Ŷ )|Z, θ}. (38.9)ŶClearly, Y ∗ (Z) in(38.9)alsodependsonθ.Using the terminology of Cressie and Wikle (2011), an empirical hierarchicalmodel (EHM) results if an estimate ˆθ(Z), or ˆθ for short, is used inplace of θ in (38.8): Inference on Y is then based on the empirical predictivedistribution,[Y |Z, ˆθ] =[Z, Y, ˆθ] × [Y |ˆθ]/[Z|ˆθ], (38.10)which means that ˆθ is also used in place of θ in (38.9).BHM inference from (38.5) and (38.6) is coherent in the sense that it emanatesfrom the well defined joint-probability distribution (38.4). However,the BHM requires specification of the prior [θ], and one often consumes largecomputing resources to obtain (38.5) and (38.6). The EHM’s inference from(38.10) can be much more computationally efficient, albeit with an empiricalpredictive distribution that has smaller variability than the BHM’s predictivedistribution (Sengupta and Cressie, 2013). Bayes’ Theorem applied to BHMor EHM for spatio-temporal data results in a typically very-high-dimensionalpredictive distribution, given by (38.6) or (38.10), respectively, whose computationrequires dimension reduction and statistical-computing algorithmssuch as EM (McLachlan and Krishnan, 2008), MCMC (Robert and Casella,2004), and INLA (Rue et al., 2009). For additional information on dimensionreduction, see, e.g., Wikle and Cressie (1999), Wikle et al. (2001), Cressie andJohannesson (2006), Banerjee et al. (2008), Cressie and Johannesson (2008),Kang and Cressie (2011), Katzfuss and Cressie (2011), Lindgren et al. (2011),and Nguyen et al. (2012).