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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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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 Spring '09
 renate
 Conditional Probability, Probability theory, posterior density, LearnBayes

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