We
first discuss the similarities and differences between classical and Bayesian
methods for a problem of learning about a population proportion.
For a particular Big Ten university, we are interested in estimating the
proportion p of athletes who graduate within six years.
For a particular year, fortyfive of seventyfour athletes admitted to
the university graduate. Assuming
that this sample is representative of athletes
We
wish to construct an interval
Suppose
that the university would like to state that over half of its athletes
graduate on time  that is, p is larger than .5.
Can the university make this statement with some confidence?
The classical 95% interval estimate for the proportion $p$ for a large sample is given by
_{
}
where
denotes the sample
proportion of athletes who graduate within six years, and n is the size of the
sample. In this example =
45/74 = .608$ and n = 74$ and, by substitution in the above formula, one obtains
the interval (.497, .719).
If the university wishes to show that the proportion p is larger than one half,
then the classical approach would test the null hypothesis H: p <= .5
against the alternative hypothesis K: p > .5.
One decides
The
Bayesian approach to learning is based on the subjective interpretation of
probability.
The prior distribution is the probability distribution that the person has before observing data. After observing data, the person changes his or her opinion about the value of the proportion. The new probability distribution, the posterior distribution, is computed using Bayes' rule. All of the person's knowledge about the proportion is contained in the posterior distribution, and statistical inferences are made by summarizing this distribution.
Let
us reanalyze our example from a Bayesian perspective. Suppose that little is known about the location of the
proportion p. We construct a prior
distribution for this proportion that reflects this belief.
After observing the graduation results, we update our probability distribution for using Bayes' rule. Let's illustrate this calculation for the single value p = .5. By Bayes' rule
Prob(p = .5 given data) is proportional to P(p = .5) x P(data given p = .5).
Here
P(p=.5)
is our prior probability of 1/99
P(data
given p = .5) is the probability of getting our data result (45 graduate out
of 74 athletes) if the true proportion is indeed equal to .5. By the
binomial formula, this probability is equal to
_{
}
So, by Bayes' rule',
_{
}
In general, the posterior probability that the proportion p is exactly equal to p0 is proportional to
Prob(p =p0 given data) is proportional to _{ }
If we perform the above calculation for each of the 99 possible values of p, and then normalize the probabilities so that they sum to one, we obtain the following posterior probability distribution. We represent it by a table and by a graph.
P 
Probability 

P 
Probability 
0.40 
0.0001


0.60

0.0702 
0.41 
0.0001


0.61

0.0709 
0.42 
0.0003


0.62

0.0694 
0.43 
0.0006


0.63

0.0658 
0.44 
0.0010


0.64

0.0604 
0.45 
0.0017


0.65

0.0536 
0.46 
0.0027


0.66

0.0459 
0.47 
0.0041


0.67

0.0380 
0.48 
0.0061


0.68

0.0303 
0.49 
0.0088


0.69

0.0233 
0.50 
0.0124


0.70

0.0172 
0.51 
0.0168


0.71

0.0122 
0.52 
0.0222


0.72

0.0082 
0.53 
0.0284


0.73

0.0053 
0.54 
0.0353


0.74

0.0033 
0.55 
0.0426


0.75

0.0019 
0.56 
0.0499


0.76

0.0010 
0.57 
0.0569


0.77

0.0005 
0.58 
0.0629


0.78

0.0002 
0.59 
0.0675


0.79

0.0001 
This probability distribution represents our current opinion about the graduating proportion of the football players.
We can estimate the unknown proportion p by some average value of the above posterior probability distribution. One reasonable estimate of p is the mode, or the most likely value. From the table above, we see that the mode is equal to .61, although the chance that p is exactly equal to .61 is only about 7%.
A
95% probability interval for the proportion is found by finding a
collection of values of p with a probability content that is approximately .95.
Here
we obtain the set {.50, .51, ...,
.71}, which is approximately equal to the 95% confidence interval found using
the classical method.
Although
the classical and Bayesian intervals agree, the interpretations of the two
intervals are
Next,
consider the question whether over half of the athletes graduate within six
years. From a Bayesian viewpoint,
the plausibility of the hypothesis H: p <= .5 is found by computing its
posterior probability. From the set
of posterior probabilities, one finds that the probability the proportion value
is less than or equal to onehalf
is .032. This probability is small,
so we would conclude that there is good evidence that over half of the athletes
graduate on time. Note that the
posterior probability of the hypothesis H is approximately equal to the
classical pvalue.
For this example, the classical procedures gave similar answers to the Bayesian
procedures when a weak or noninformative prior distribution was used.
However, there are important distinctions between the two sets of
procedures. One difference is the
interpretation. The Bayesian
computes probabilities about the unknown proportion
The Bayesian mode of inference has a number of desirable features.
Inferential
statements are easy to understand
One attractive feature is that inferential statements about a parameter are easy to communicate. It is natural to talk about the probability that p falls in an interval or the probability that a hypothesis is true.
One recipe
A second nice feature is that a single recipe, Bayes' rule, is used for updating one's probabilities about a parameter. This rule can be used for small or large sample sizes.
Mechanism
for using subjective beliefs in a problem
Bayesian
methods allow a person to use his or her subjective beliefs about the location
of the parameter in the inference problem.
In our example, one may have some opinions about graduating rates of
athletes based on data from other universities, and a prior probability
distribution for p can be constructed to reflect this knowledge.
Bayes' rule provides a useful mechanism for combining this prior
knowledge about the graduation rates with information contained in the sample.