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The last expression may be recognized as a beta density (see (34), 4) with a 4 and b 2. Since the normal1 5! izing constant here should be (see Appendix A), we deduce that the constant of proportionality is 20 and B(4, 2) 3!1! 3(1 p(uu x) 20u u), 0 u 1. The graphs of the prior (uniform) and posterior densities are shown in Fig. 11-1. The mean and variance are, respectively, 0.5 and 1 > 12 < 0.08 for the prior density whereas they are 2 > 3 < 0.67 and 8 > 252 < 0.03 for the posterior density. The shift to the right and the increased concentration about the mean as we move from the prior to the posterior density are evident in Fig. 11-1.
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Fig. 11-1
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The result obtained in Example 11.5 may be generalized in a straightforward manner. Suppose that X has a binomial distribution with parameters n and u (see (1), 4) and that the prior probability distribution of u is beta with density function (see (34), 4): p(u) ua 1(1 u)b B(a, b)
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where B(a, b) is the beta function (see Appendix A). (Note that if a b 1, then p(u) is the uniform density on [0, 1] the situation discussed in Example 11.5.) Then the posterior density p(u ux) corresponding to any observed value x is given by p(u u x) f (x u u)p(u) 3 f (x u u)p(u) dp ~ ux(1 u)n
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ux a 1(1 u)n x b 1 B(x a, n x b)
CHAPTER 11 Bayesian Methods
a and n x b. We thus have the following:
This may be recognized as a beta density with parameters x Theorem 11-1
If X is a binomial random variable with parameters n and u, and the prior density of u is beta with parameters a and b, then the posterior density of u after observing X x is beta with parameters x a and n x b.
EXAMPLE 11.6 Suppose that X is binomial with parameters n 10 and unknown u and that p(u) is beta with parameters a b 2. If an observation on X yielded x 2, then the posterior density p(u u x) may be determined as follows. From Theorem 11-1 we see that p(u u x) is beta with parameters 4 and 10. The prior (symmetric about 0.5) and posterior densities are shown in Fig. 11-2. It is clear that the effect of the observation on the prior density of u is to shift its mean from 0.5 down to 4>14 < 0.29 and to shrink the variance from 0.05 down to 0.014 (see (36), 4).
3 2.5 2 1.5 1 0.5
0.4
Fig. 11-2
Sampling From a Poisson Population
Theorem 11-2 If X is a Poisson random variable with parameter l and the prior density of l is gamma with parameters a and b (as in (31), 4), then the posterior density of l, given the sample x1, x2, c, xn, is gamma with parameters nx a and b>(1 nb), where x is the sample mean. # # If x1, x2, c, xn is a sample of n observations on X, then the likelihood of x ln x f (xZu) e nl x !x !cx ! . We are given the prior density of l: 1 2 n p(l) It follows that the posterior density of l is p(lu x) f (xul)p(l) 3 f (xul)p(l) dl ~ e
` l nx 3e l la 1e 0 l>bdl nlln x
(x1, x2, c, xn) may be written as
la 1e l>b ba (a)
la 1e
nb)n x bn x
a (n x a) 1 l e l(nb 1)>b a
(nx #
The last expression may be recognized as a gamma density, thus proving Theorem 11-2.
EXAMPLE 11.7 The number of defects in a 1000-foot spool of yarn manufactured by a machine has a Poisson distribution with unknown mean l. The prior distribution of l is gamma with parameters a 3 and b 1. A total of eight defects were found in a sample of five spools that were examined. The posterior distribution of l is gamma with parameters a 11 and b 1>6 < 0.17. The prior mean and variance are both 3 while the posterior mean and variance are respectively 1.87 and 0.32. The two densities are shown in Fig. 11-3.
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