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CHAPTER 11 Bayesian Methods
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Theorem 11-8 The median of t(u) with respect to the posterior distribution p(uu X) is the Bayes estimator of t(u) for the absolute error loss function L(u, a) u u au . When t(u) u, these two theorems reduce to the optimality results mentioned earlier for the posterior mean and median as estimates of u.
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EXAMPLE 11.15 Suppose X is a binomial random variable with parameters n and u, and the prior density of u is beta with parameters a b 1. Theorems 11-7 and 11-8 may then be used to obtain the Bayes estimates of u(1 u) for the (a) squared error and (b) absolute error loss functions.
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(a) We obtain the posterior mean of u(1 E(u(1 u)u x) E(u u x)
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u) from Theorem 11-1. We have E(u2 u x) x n 1 2 B (x (n 1)(x 2)(n
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(b) The median of the posterior distribution of u(1 u) may be obtained numerically from the posterior distribution of u using computer software. To show the work involved, let us assume that n 10 and x 4. The posterior distribution of u is beta with parameters 5 and 7. The median of u(1 u), say m, satisfies the condition P(u(1 u) !1 4m !1 4m 1 1 m) 0.5, which is equivalent to the requirement that PQu R PQu R 0.5 2 2 2 2 under the beta distribution with parameters 5 and 7. The solution is m 0.247. (The posterior mean of u(1 u) in this case is 0.224.)
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1)(n x 1) 2)(n 3)
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Bayesian Interval Estimation
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Given the posterior density function p(uu x) for a parameter u, any u
interval [uL, uU] with the property
3 p(uu x) du
(14)
is called a Bayesian (1 a) 100% credibility interval for u. Of the various possible intervals that satisfy this property, two deserve special mention: the equal tail area interval and the highest posterior density (HPD) interval. The equal tail area (1 a) 100% interval has the property that the area under the posterior density to the left of uL equals the area to the right of uU:
uL `
3 p(u u x) du
3 p(uu x) du
a)>2
The requirement for the HPD interval is that, in addition to (14), we have p(uu x) p(ur u x) if u H [uL, uU] and ur x [uL, uU]. Clearly if p(u ux) does not have a unique mode, then the set of u values satisfying the last condition may not be an interval. To avoid this possibility, we shall assume here that the posterior density is unimodal. It follows directly from this assumption that p(uL u x) p(uU ux) and that for any a the HPD interval is the shortest of all possible (1 a) 100% credibility intervals. But equal tail area intervals are much easier to construct from the readily available percentiles of most common distributions. The two intervals coincide when the posterior density is symmetric and unimodal.
EXAMPLE 11.16 Suppose that a random sample of size 9 from a normal distribution with unknown mean u and variance 1 yields a sample mean of 2.5. Also suppose that the prior distribution of u is normal with mean 0 and variance 1. From Theorem 11-3, we see that the posterior distribution of u is normal with mean 2.25 and variance 0.1. A 95% equal tail credibility interval for u is given by [uL, uU] with uL and uU equal, respectively, to the 2.5th percentile and 97.5th percentile of the normal density with mean 2.25 and variance 0.1. From Appendix C, we then have uL < 2.25 (2.36 0.32) 1.49 and uU < 2.25 (2.36 0.32) 3.01. The 95% Bayesian equal tail credibility interval (and the HPD interval, because of the symmetry of the normal density) is thus given by [1.49, 3.01]. EXAMPLE 11.17 In Problem 6.6, we obtained traditional confidence intervals for a normal mean u based on a sample of size n 200 assuming that the population standard deviation was s 0.042. The 95% confidence interval for