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Table 11-2 u p(u u D) 0.2 32>157 0.5 125>157
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It is convenient at this point to introduce some notation that is particularly helpful for presenting Bayesian methods. Suppose that X is a random variable with probability or density function f (x) that depends on an unknown parameter u. We assume that our uncertainty as to the value of u may be represented by the probability or density function p(u) of a random variable . The function f (x) may then be thought of as the conditional u; we shall therefore denote f (x) by f (x Z u) throughout this probability or density function of X given chapter. Also, we shall denote the joint probability or density function of X and by f (x; u) f (x Z u) p(u) and the posterior (or conditional) probability or density function of given X x by p(uu x). If x1, x2, c, xn is a random sample of values of X, then the joint density function of the sample (also known as the likelihood
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CHAPTER 11 Bayesian Methods
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function, see (19), 6) will be written using the vector notation x (x1, x2, c, xn) as f (xZu) f (x1 u u) f (x2 u u) c f (xn u u); similarly, the posterior probability or density function of u given the sample will be denoted by p(u u x). The following version of Bayes theorem for random variables is a direct consequence of (26) and (43), 2: p(u u x) f (x; u) f (x) f (xu u)p(u) 3 f (x u u)p(u) du (1)
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where the integral is over the range of values of and is replaced with a sum if is discrete. In our applications of Bayes theorem, we seldom have to perform the integration (or summation) appearing in the denominator of (1) since its value is independent of u. We can therefore write (1) in the form p(u u x) ~ f (xu u)p(u) (2)
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meaning that p(u u x) C f (x u u)p(u), where C is a proportionality constant that is free of u. Once the functional form of the posterior density is known, the normalizing constant C can be determined so as to make p(u u x) a probability density function. (See Example 11.4.) Remark 1 The convention of using upper case letters for random variables is often ignored in Bayesian presentations when dealing with parameters, and we shall follow this practice in the sequel. For instance, in the next example, we use l to denote both the random parameter (rather than ) and its possible values.
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EXAMPLE 11.4 The random variable X has a Poisson distribution with an unknown parameter l. It has been determined that l has the subjective prior probability function given in Table 11-3. A random sample of size 3 yields the X-values 2, 0, and 3. We wish to find the posterior distribution of l.
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Table 11-3 l p(l) 0.5 1>2 1.0 1>3 1.5 1>6
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The likelihood of the data is f (xu l)
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lx1 x2 x3 . From (1) and (2), we have the posterior density x1!x2!x3!
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3llx1 x2 x3p(l)
p(lu x)
x1!x2!x3! 1 e 3llx1 x1!x2!x3! a l
x2 x3p(l)
3ll5p(l)
0.5, 1, 1.5
The constant of proportionality in the preceding is simply the reciprocal of the sum g l e 3ll5p(l) over the three possible values of l. By substituting l 0.5, 1.0, 1.5, respectively, and p(l) from Table 11-3 into the preceding sum, and then normalizing so that the sum of the probabilities p(l ux) is equal to 1, we obtain the values in Table 11-4. Table 11-4 l p(l u x) 0.5 0.10 n ux(1 x 1.0 0.49 1.5 0.41
EXAMPLE 11.5 The random variable X has a binomial distribution with probability function given by
f (x Zu)
1, 2, c, n
CHAPTER 11 Bayesian Methods
Suppose that nothing is known about the parameter u so that a uniform (vague) prior distribution on the interval [0, 1] is chosen for u. If a sample of size 4 yielded 3 successes, then the posterior probability density function of u may be obtained using (2): 4 u3(1 3 u) 1 ~ u3(1 u) du u)
p(u u x)