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3.2. SAMPLING TO SUMMARIZE 69Density0.000 0.001 0.002 0.003 0.004meanmedianmode0.0 0.2 0.4 0.6 0.8 1.0proportion water (p)expected proportional loss0.2 0.4 0.6 0.80.0 0.2 0.4 0.6 0.8 1.0decisionFIGURE 3.4. Point estimates and loss functions. Le: Posterior distribution(blue) aer observing 3 water in 3 tosses of the globe. Vertical lines showthe locations of the mode, median, and mean. Each point implies a differentloss function. Right: Expected loss under the rule that loss is proportionalto absolute distance of decision (horizontal axis) from the true value. epoint marks the value of p that minimizes the expected loss, the posteriormedian.mean( samples )median( samples )R code3.16[1] 0.8005558[1] 0.8408408ese are also point estimates, and they also summarize the posterior. But all three—themode (MAP), mean, and median—are different in this case. How can we choose amongthem? FIGURE 3.4 shows this posterior distribution and the locations of these point summaries.One principled way to go beyond using the entire posterior as the estimate is to choosea LOSS FUNCTION. A loss function is a rule that tells you the cost associated with using anyparticular point estimate. While statisticians and game theorists have long been interestedin loss functions, and how Bayesian inference supports them, scientists hardly ever use themexplicitly. e key insight is that different loss functions imply different point estimates.Here’s an example to help us work through the procedure. Suppose I offer you a bet.Tell me which value of p, the proportion of water on the Earth, you think is correct. I willpay you one-hundred United States Dollars, if you get it exactly right. But I will subtractmoney from your gain, proportional to the distance of your decision from the correct value.Precisely, your loss is proportional to the absolute value of d−p, where d is your decision andp is the correct answer. We could change the precise dollar values involved, without changingthe important aspects of this problem. What matters is that the loss is proportional to thedistance of your decision from the true value.