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An ad hoc rule sometimes used is to reject the null hypothesis if BF 1. It can be shown that this is equivalent to the Bayes a test with a P(H0) : Reject H0 if P(H0 u x) P(H0).
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EXAMPLE 11.22 Let us determine the rejection criterion in terms of the Bayes factor for the test used in Example 11.19. We have a 0.05, and P(H0) P(u 0.3) 0.5 . Therefore, by Theorem 11-10, the test criterion is to reject (0.05)(0.5) H0 if BF < 0.053. The Bayes factor corresponding to 16 successes out of 30 trials is (0.95)(0.5) P(H0 u x) P(H0) 0.5 0.037 b^a a b a b a b < 0.038. Since this is less than 0.053, we reject the null hypothesis. P(H1) 1 0.037 ^ 0.5 P(H1 u x) EXAMPLE 11.23 In Example 11.18, suppose we wish to employ the decision rule to reject H0 if the Bayes factor is less than 1. We already know that the probability of the null hypothesis under the posterior density of u is 0.20. 0.20 1 The posterior odds for H0 are therefore . The prior probability for H0 is given by P(u 0.3) 0.80 4
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0.39 0.3 0.4 . The Bayes factor for H0 is (1>4)>(39>61) < 0.39 b < 0.39, so the prior odds for H0 are 0.36 0.61 decision therefore is to reject H0.
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CHAPTER 11 Bayesian Methods
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The Bayesian framework makes it possible to obtain the conditional distribution of future observations on the basis of a currently available prior or posterior distribution for the population parameter. These are known as predictive distributions and the basic process involved in their derivation is straightforward marginalization of the joint distribution of the future observations and the parameter (see pages 40 41). Suppose that n Bernoulli trials with unknown success probability u result in x successes and that the prior density of u is beta with parameters a and b. If m further trials are contemplated on the same Bernoulli population, what can we say about the number of successes obtained We know from Theorem 11-1 that the posterior distribution of u, given x, is beta with parameters x a and n x b. If f (y u u) is the probability function of the number Y of successes in the m future trials, the joint density of Y and u is f ( y, u u x) f ( y u u)p(uu x) m ux y a 1(1 u)m n B(x a, n x y m uy(1 y u)m
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ux a 1(1 u)n x b 1 B(x a, n x b) 0 u 1, y 0, 1, c, m
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x y b 1
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The predictive probability function of Y, denoted by f *(y), is the marginal density of Y obtained from the above joint density by integrating out u: f
*( y)
m ux y a 1(1 u)n m 3 y B(x a, n x
x y b 1
We thus have the following theorem.
Theorem 11-11 If n Bernoulli trials with unknown success probability u result in x successes, and the prior density of u is beta with parameters a and b, then the predictive density of the number of successes Y in m future trials on the same Bernoulli population is given by (20). Remark 5 It is evident from (19) that f *( y) may also be regarded as the expectation, E ( f (y u u)), of the probability function of Y with respect to the posterior density p(uu x) of u.
EXAMPLE 11.24 Suppose that 7 successes were obtained in 10 Bernoulli trials with success probability u. An independent set of 8 more Bernoulli trials with the same success probability is being contemplated. What could be said about the number of future successes if u has a uniform prior density in the interval [0, 1]
m B(x y
du b)
(19)
y a, m n x y B(x a, n x b)