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a 3 3p(uu x)PII (u) dx du
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EXAMPLE 11.18 Suppose that the reaction time (in seconds) of an individual to certain stimuli is known to be normally distributed with unknown mean u but a known standard deviation of 0.30 sec. The prior density of u is normal with m 0.4 sec and y2 0.13. A sample of 20 observations yielded a mean reaction time of 0.35 sec. We wish to test the null hypothesis H0 : u 0.3 against the alternative H1 : u 0.3 using a Bayes 0.05 test. By Theorem 11-3, the posterior density is normal with mean 0.352 and variance 0.004. The posterior probability of 0.3 0.352 H0 is therefore given by P(u 0.3) P A Z B < 0.20. Since this probability is greater than 0.05, we can0.063 not reject H0. EXAMPLE 11.19 X is a Bernoulli random variable with success probability u, which is known to be either 0.3 or 0.6. It is desired to test the null hypothesis H0 : u 0.3 against the alternative H1 : u 0.6 using a Bayes 0.05 test assuming the vague prior probability distribution for u : P(u 0.3) P(u 0.6) 0.5. A sample of 30 trials on X yields 16 successes. To check the rejection criterion of the Bayes 0.05 test, we need the posterior probability of the null hypothesis:
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P(x 16 u u 0.3) P(u 0.3) 0.3) P(u 0.3) P(x 16 u u 0.6) P(u
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(0.0042)(0.5) < 0.037 (0.0042)(0.5) (0.1101)(0.5) Since this probability is less than 0.05, we reject the null hypothesis.
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When the prior distribution involved is proper, Bayesian statistical inference can be formulated in the language of odds (see page 5) using what are known as Bayes factors. Bayes factors may be regarded as the Bayesian analogues of likelihood ratios on which most of the classical tests in 7 are based. Consider the hypothesis testing problem discussed in the previous section. We are interested in testing the null hypothesis H0 : u H 0 against the alternative hypothesis H1 : u H r0. The quantities 3p(u) du P(H0) P(H1)
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3 p(uu x) du and P(H0 u x) P(H1 u x)
(17) 3 p(uu x) du
3 p(u) du
are known respectively as the prior and posterior odds ratios of H0 relative to H1. The Bayes factor (BF for short) is defined as the posterior odds ratio over the prior odds ratio. Using the fact that p(uu x)~f (xu u)p(u), we can write the Bayes factor in the following form: P(H0 u x) P(H0) ^ P(H1) P(H1 u x) 1 f P(H0) 3 (xu u)p(u) du
Posterior odds ratio Prior odds ratio
1 f (xu u)p(u) du P(H1) 3
(18)
CHAPTER 11 Bayesian Methods
The Bayes factor is thus the ratio of the marginals (or the averages) of the likelihood under the two hypotheses. It can also be seen from (18) that when the hypotheses are both simple, say H0 : u u0 and H1 : u u1 , the Bayes factor becomes the familiar likelihood ratio of classical inference: BF
EXAMPLE 11.20 alternative, H1 : u
f (xu u0) . f (xu u1)
In Example 11.18, let us calculate the Bayes factor for the null hypothesis H0 : u 0.3 against the 0.3, using (18). We need P(H0) P(u 0.3), where u is a normal random variable with mean 0.4
0.3 0.4 b < 0.39. The posterior probability of the null hypothesis, available 0.36 P(H0 u x) P(H0) 0.39 0.20 b^a b a b a b < 0.39. from Example 11.18, is P(H0 u x) < 0.20. The Bayes factor is a P(H1) 0.80 ^ 0.61 P(H1 u x) and variance 0.13. This equals PaZ
EXAMPLE 11.21 A box contains a fair coin and two biased coins (each with P( heads ) 0.2). A coin is randomly chosen from the box and tossed 10 times. If 4 heads are obtained, what is the Bayes factor for the null hypothesis H0 that the chosen coin is fair relative to the alternative H1 that it is biased The prior probabilities are P(H0) 1 > 3 and (0.5)10 P(H1) 2>3, so the prior odds ratio is 0.5. The posterior probabilities are P(H0 u x) < 0.54 (0.5)10 2(0.2)4(0.8)6 (0.2)4(0.8)6 < 0.46 , so the posterior odds ratio is 0.54 > 0.56 < 1.16. The Bayes factor is and P(H1 u x) 10 (0.5) (0.2)4(0.8)6
therefore 1.16>0.5 < 3.32. We can also get the same result directly as the ratio of the likelihoods under the two hy10 10 a b(0.5)10 and P(xuH1) a b(0.2)4(0.8)6. potheses P(xuH0) 4 4
It can be seen from (18) that the Bayes factor quantifies the strength of evidence afforded by the data for or against the null hypothesis relative to the alternative hypothesis. Generally speaking, we could say that if the Bayes factor is larger than 1, the observed data adds confirmation to the null hypothesis and if it is less than 1, then the data disconfirms the null hypothesis. Furthermore, the larger the Bayes factor, the stronger the evidence in favor of the null hypothesis. The calibration of the Bayes factor to reflect the actual strength of evidence for or against the null hypothesis is a topic that will not be discussed here. We can, however, prove the following theorem: Theorem 11-10 The Bayes a test is equivalent to the test that rejects the null hypothesis if BF a[1 P(H0)] . (1 a)P(H0)
To see this, note that the rejection criterion of a Bayes a test, namely P(H0 u x) a, is equivalent to the cona[1 P(H0)] P(H0 u x) a dition and that this inequality is equivalent to the condition BF . 1 a (1 a)P(H0) P(H1 u x)