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76 3. SAMPLING THE IMAGINARY0.38 (A) p = 0.38number 0.89 (C) p = of 0.89 water samplesFrequency0 1000 3000Frequency0 1000 30000 3 6 9number 0.64 (B) p = of 0.64 water samplesMergedprobability0.000 0.010 0.020A BCFrequency0 1000 3000Frequency0 1000 30000 0.5 10 3 6 90 3 6 9probability of waternumber of water samples0 3 6 9number of water samplesFIGURE 3.6. Simulating predictions from the total posterior. Le: e familiarposterior distribution for the globe tossing data. ree example parametervalues (0.38, 0.64, 0.89) are marked by the vertical lines. Middlecolumn: Each of the three parameter values is used to simulate observations.Right: Combining simulated observation distributions for all parametervalues (not just 0.38, 0.64, and 0.89), each weighted by its posteriorprobability, produces the posterior predictive distribution. is distributionpropagates uncertainty about parameter to uncertainty about prediction.Observed value (6) highlighted.can compute intervals and point statistics using the same procedures. If you plot these samples,you’ll see the distribution shown in the right-hand plot in FIGURE 3.6.e simulated model predictions are quite consistent with the observed data in thiscase—the actual count of 6 lies right in the middle of the simulated distribution. ere isquite a lot of spread to the predictions, but a lot of this spread arises from the binomial processitself, not uncertainty about p. Still, it’d be premature to conclude that the model isperfect. So far, we’ve only viewed the data just as the model views it: each toss of the globe iscompletely independent of the others. is assumption is questionable. Unless the persontossing the globe is careful, it is easy to induce correlations and therefore patterns amongthe sequential tosses. Consider for example that about half of the globe (and planet) is coveredby the Pacific Ocean. As a result, water and land are not uniformly distributed on theglobe, and therefore unless the globe spins and rotates enough while in the air, the positionwhen tossed could easily influence the sample once it lands. e same problem arises in coin