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N. Cressie 433In the last 20 years, methodological research in Statistics has seen a shiftfrom mathematical statistics towards statistical computing. Deriving an analyticalform for (38.6) or (38.10) is almost never possible, but being able tosample realizations from them often is. This shift in emphasis has enormouspotential for EI.For economy of exposition, I feature the BHM in the following discussion.First, if I can sample from the posterior distribution, [Y,θ|Z], I can automaticallysample from the predictive distribution, [Y |Z], by simply ignoring theθ’s in the posterior sample of (Y,θ). This is called a marginalization propertyof sampling. Now suppose there is scientific interest in a summary g(Y ) of Y(e.g., regional averages, or regional extremes). Then an equivariance propertyof sampling implies that samples from [g(Y )|Z] are obtained by sampling from[Y |Z] and simple evaluating each member of the sample at g. Thisequivarianceproperty is enormously powerful, even more so when the sampling doesnot require knowledge of the normalizing term [Z] in(38.5).Thebestknownstatistical computing algorithm that samples from the posterior and predictivedistributions is MCMC; see, e.g., Robert and Casella (2004).Which summary of the predictive distribution [g(Y )|Z] willbeusedtoestimate the scientifically interesting quantity g(Y )? Too often, the posteriormean,∫E{g(Y )|Z} = g(Y )[Y |Z]dY,is chosen as a “convenient” estimator of g(Y ). This is an optimal estimatorwhen the loss function is squared-error: L{g(Y ), ĝ} = {ĝ − g(Y )} 2 ;see,e.g.,Berger (1985). However, squared-error loss assumes equal consequences (i.e.,loss) for under-estimation as for over-estimation. When a science or policyquestion is about extreme events, the squared-error loss function is strikinglyinadequate, yet scientific inference based on the posterior mean is ubiquitous.Even if squared-error loss were appropriate, it would be incorrect to computeE(Y |Z) and produce g{E(Y |Z)} as an optimal estimate, unless g is alinear functional of Y . However, this is also common in the scientific literature.Under squared-error loss, the optimal estimate is E{g(Y )|Z}, whichisdefined above. Notice that aggregating over parts of Y defines a linear functionalg, but that taking extrema over parts of Y results in a highly non-linearfunctional g. Consequently, the supremum/infimum of the optimal estimateof Y (i.e., g{E(Y |Z)}) is a severe under-estimate/over-estimate of the supremum/infimumof Y ,i.e.,g(Y ).38.4 Smoothing the dataEI is fundamentally linked to environmental data and the questions that resultedin their collection. Questions are asked of the scientific process Y , and