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CHAPTER 11 Bayesian Methods
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the population mean came out to be [0.82, 0.83]. It is instructive now to obtain the actual posterior probability for this interval obtained assuming normal prior distribution for u with mean m 1 and standard deviation y 0.05. From Theorem 11-3, we see that the posterior density has mean mpost < 0.825 and standard deviation ypost < 0.003. The area under this density over the interval [0.82, 0.83] is 0.9449.
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A basic conceptual difference between conventional confidence intervals and Bayesian credibility intervals should be pointed out. The confidence statement associated with a 100 a% confidence interval for a parameter u is the probability statement PX(L(X) u U(X)) a in the sample space of observations, with the frequency interpretation that in repeated sampling the random interval [L(X), U(X)] will enclose the constant u (x1, x2, c, xn) of observations on X, the statement 100 a% of the times. But, given a random sample x P(L(x) u U(x)) a (in words, we are 100 a% sure that u lies between L(x) and U(x) ) is devoid of any sense simply because u, L(x), and U(x) are all constants. The credibility statement associated with a Bayesian 100 a% credibility interval is the probability statement P (L(x) u U(x)) a in the parameter space endowed with the probability density p(uu x). Although this statement may not have a frequency interpretation, it nonetheless is a valid and useful summary description of the distribution of the parameter to the effect that the interval [L(x), U(x)] carries a probability of a under the posterior density p(uu x).
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Bayesian Hypothesis Tests
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Suppose we wish to test the null hypothesis H0 : u u0 against the alternative hypothesis H1 : u u0. Then a reasonable rule for rejecting H0 in favor of H1 could be based on the posterior probability of the null hypothesis given the data,
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P(H0 u x)
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3 p(uu x) du
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(15) a. A test based
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For instance, we could specify an a 0 and decide to reject H0 whenever x is such that P(H0 u x) on this rejection criterion is known as a Bayes a test.
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Remark 3 The Bayesian posterior probability of the null hypothesis shown in (15) is quite different from the P value of a test (see page 215) although the two are frequently confused for each other, and the latter is often loosely referred to as the probability of the null hypothesis. We now show an optimality property enjoyed by Bayes a tests. We saw in 7 that the quantities of primary interest in assessing the performance of a test are the probabilities of Type I error and Type II error for each u. If C is the critical region for a test, then these two probabilities are given by PI(u) c 3C 0 f (x, u) dx, u u u0 u0 and PII(u) c 3Cr 0 f (x, u)dx, u u u0 u0
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For any specified a, the following weighted mean of these two probabilities is known as the Bayes risk of the test.
u0 `
r(C)
a) 3 3p(u u x)PI (u) dx du
a 3 3p(uu x)PII (u) dx du
u0 Cr
(16)
For each fixed x, the quantity on the right may be written as (1 a)P(u aP(u u0 ux)IC (x) u0 u x) [(1 aP(u a)P(u u0 u x)ICr (x) u0 u x)IC (x) (1 aP(u a)P(u u0 ux)IC (x) aP(u u0 ux)(1 ICr (x)) u0 u x)IC (x)]
where IE (x) denotes the indicator function of the set E. The term inside brackets is minimized when the critical region C is defined so that IC (x) e 1 0 if (1 a)P(u otherwise u0 ux) aP(u u0 ux)
CHAPTER 11 Bayesian Methods
This shows that r(C) is minimized when C consists of those data points x for which P(u u0 u x) a. We have thus established that the Bayes a test minimizes the Bayes risk defined by (16). In general, we have the following theorem. Theorem 11-9 For any subset 0 of the parameter space, among all tests of the null hypothesis H0 : u H 0 against the alternative H1 : u H r0, the Bayes a test, which rejects H0 if P(u H 0 u x) a minimizes the Bayes risk defined by r(C) (1 a) 3 3p(u u x)PI (u) dx du