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2 Introduction to Bayesian Thinking 2.1 Introduction In this chapter, the basic elements of the Bayesian inferential approach are introduced through the basic problem of learning about a population propor- tion. Before taking data, one has beliefs about the value of the proportion and one models his or her beliefs in terms of a prior distribution. We will illus- trate the use of diﬀerent functional forms for this prior. After data have been observed, one updates one’s beliefs about the proportion by the computation of the posterior distribution. One summarizes this probability distribution to perform inferences. Also one may be interested in predicting the likely out- comes of a new sample taken from the population. Many of the commands in the R base package can be used in this setting. The probability distribution commands such as dbinom and dbeta and simu- lation commands such as rbeta , rbinom and sample are helpful in simulating draws from the posterior and predictive distributions. Also we illustrate some special R commands pdisc , histprior ,and discint in the LearnBayes pack- age that are helpful in constructing priors and computing and summarizing a posterior. 2.2 Learning About the Proportion of Heavy Sleepers Suppose a person is interested in learning about the sleeping habits of Amer- ican college students. She hears that doctors recommend eight hours of sleep for an average adult. What proportion of college students get at least eight hours of sleep? Here we think of a population consisting of all American college students and let p represent the proportion of this population who sleep (on a typical night during the week) at least eight hours. We are interested in learning about the location of p .

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20 2 Introduction to Bayesian Thinking The value of the proportion p is unknown. In the Bayesian viewpoint a person’s beliefs about the uncertainty in this proportion are represented by a probability distribution placed on this parameter. This distribution reﬂects the person’s subjective prior opinion about plausible values of p . A random sample of students from a particular university will be taken to learn about this proportion. But ﬁrst the person does some initial research to learn about the sleeping habits of college students. This research will help her in constructing a prior distribution. In the Internet article “College Students Don’t Get Enough Sleep” in The Gamecock , the student newspaper of the University of South Carolina (April 20, 2004), the person reads that a sample survey reports that most students spend only six hours sleeping. She reads a second article “Sleep on It: Imple- menting a Relaxation Program into the College Curriculum”in Fresh Writing , a 2003 publication of the University of Notre Dame. Based on a sample of 100 students, “approximately 70% reported receiving only ﬁve to six hours of sleep on the weekdays, 28% receiving seven to eight, and only 2% receiving the healthy nine hours for teenagers.”
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