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3.2. SAMPLING TO SUMMARIZE 65Density0.0000 0.0010 0.00200.00 0.25 0.50 0.75 1.00proportion water (p)Density0.0000 0.0010 0.00200.00 0.25 0.50 0.75 1.00proportion water (p)lower 80%middle 80%Density0.0000 0.0010 0.00200.00 0.25 0.50 0.75 1.00proportion water (p)Density0.0000 0.0010 0.00200.00 0.25 0.50 0.75 1.00proportion water (p)FIGURE 3.2. Two kinds of confidence interval. Top row: intervals of definedboundaries. Top-le: the blue area is the posterior probability below a parametervalue of 0.5. Top-right: the posterior probability between 0.5 and0.75. Bottom row: intervals of defined mass. Bottom-le: lower 80% posteriorprobability exists below a parameter value of about 0.75. Bottom-right:middle 80% posterior probability lies between the 10% and 90% quantiles.quantile( samples , c( 0.1 , 0.9 ) )R code3.1010% 90%0.4464464 0.8118118is region is shown in the bottom-right plot in FIGURE 3.2.Intervals of this sort, which assign equal probability mass to each tail, are very commonin the scientific literature. We’ll call them PERCENTILE INTERVALS (PI). ese intervals doa good job of communicating the shape of a distribution, as long as the distribution isn’t tooasymmetrical. But in terms of supporting inferences about which parameters are consistentwith the data, they are not perfect. Consider the posterior distirbution and different intervalsin FIGURE 3.3. is posterior is consistent with observing 3 waters in 3 tosses and a uniform