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CHAPTER 11 Bayesian Methods
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When a b 1 in Theorem 11-5, the posterior mode estimate gpost of u reduces to the maximum likelihood estimate x>n. This was also pointed out in Example 11.13 but the result is obviously not true for general a and b. But, regardless of the values of a and b, when the sample size is large enough, both mpost and gpost will be close to the sample proportion x>n. Furthermore, for all n, mpost is a convex combination of the prior mean of u and the sample proportion. (See Problem 11.38.)
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EXAMPLE 11.14 Suppose that a random sample of size n is drawn from a normal distribution with unknown mean u and variance 1. 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 nx >(1 n). # Clearly, the posterior mean, median, and mode are identical here and would therefore lead to the same point estimate, nx >(1 n), of u. It was shown in Problem 6.25, page 206, that the maximum likelihood estimate for u in this case is the # sample mean x, which is known to be unbiased (Theorem 5-1). On the other hand, the Bayesian estimates derived here # are biased, although they are asymptotically unbiased.
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A general result along these lines follows easily from Theorem 11-3 and is as follows. Theorem 11-6 Suppose that a random sample of size n is drawn from a normal distribution with unknown mean u and known variance s2. Also suppose that the prior distribution of u is normal with mean m and variance y2. Then the posterior mean, median, and mode all provide the same estimate of u, namely (s2m ny2 x)>(s2 ny2), where x is the sample mean. # #
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As we saw in the binomial case, the posterior mean estimate mpost just obtained lies between the prior mean m and the maximum likelihood estimate x of u. This may be seen by writing mpost in the form # [s2 >(s2 ny2)] m [ny2 >(s2 ny2)] x, as a convex combination of the two. We can also see from this ex# pression that for large n, mpost will be close to x and will not be appreciably influenced by the prior mean m. # An optimality property of mpost as an estimate of u directly follows from Theorem 3-6. Indeed, we can prove a more general result along these lines using this theorem. Suppose we are interested in estimating a function of u, say t(u). For any set of observations x from f (x Zu), if we define the statistic T(x) as the posterior expectation of t(u), namely
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T(x) then it follows from Theorem 3-6 that
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E(t(u)u x)
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3 t(u)p(uux) du
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E[(t(u) is a minimum when a(x)
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3 (t(u)
a(x))2p(uu x) du
T(x). In other words, T(x) satisfies the property E[(t(u) T(x))2 u x] min E[(t(u) a a(x))2 u x] for each x (12)
since T(x) is the mean of t(u) with respect to the posterior density p(u u x). In the general theory of Bayesian estimation, we typically start with a loss function L(t(u), a) that measures the distance between the parameter and an estimate. We then seek an estimator, say d*(X), with the property that E[L(t(u), d*(x))u x] min E[L(t(u), a(x))u x] for each value x of X a (13)
where the expectation is over the parameter space endowed with the posterior density. An estimator satisfying equation (13) is called a Bayes estimator of t(u) with respect to the loss function L(t(u), a). The following theorem then is just a restatement of (12): Theorem 11-7 The mean of t(u) with respect to the posterior distribution p(u u X) is the Bayes estimator of t(u) for the squared error loss function L(u, a) (u a)2. Another common loss function is the absolute error loss function L(u, a) u u au. It is shown in Problem 11.100 that the median of the posterior density is the Bayes estimator for this loss function.