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(20)
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The predictive distribution of the number of future successes may be obtained from (20) with a 10, m 8, and x 7:
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f *( y) 8 B( y 8, 12 B(8, 4) y 3 0.089 y) y 0, 1, c, 8
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Table 11-5 summarizes the numerical results.
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Table 11-5 4 0.153 5 0.210 6 0.227 7 0.182 8 0.085
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Remark 6 In an earlier remark, following Theorem 11-5, on Laplace s law of succession, it was pointed out that if all n trials of a binomial experiment resulted in successes, then the probability that a future trial will also result in success may be estimated by the posterior mean of the success parameter u, namely (n 1)>(n 2). The same result can be obtained as a special case of (20) with a b 1, m 1, and x n. The predictive distribution of a future observation turns out to be binomial with success probability (n 1)>(n 2). The two approaches, however, do not lead to the same results beyond n 1. For instance, if we take the posterior mean (n 1)>(n 2) as the success probability for the m future trials, then the probability that all of them are successes would be [(n 1)>(n 2)]m, but (20) gives us (n 1)>(m n 1).
CHAPTER 11 Bayesian Methods
EXAMPLE 11.25 In Example 11.24, suppose that we are interested in predicting the outcome of the first 10 Bernoulli trials before they are performed. Determine the predictive distribution of the number of successes, say X, in the 10 trials, again assuming that u has a uniform prior density in the interval [0, 1]. The joint distribution of X and u is given by
f (x; u)
The marginal density of x may be obtained from this by integrating out u: f * (x) 10 x 3 x u (1
10 x u (1 x
u)10
0, 1, c, 10
u)10 x du
Remark 7 The predictive distributions obtained in Examples 11.24 and 11.25 are different in that they are based, respectively, on a posterior and a prior distribution of the parameter. A distinction between prior predictive distributions and posterior predictive distributions is sometimes made to indicate the nature of the parameter distribution used. Predictive distributions for future normal samples may be derived analogously. We saw in Theorem 11-3 that if we have a sample of size n from a normal distribution with unknown mean u and known variance s2 and if u is normal with mean m and variance y2, then the posterior distribution of u is also normal, with mean mpost and variance y2 given by (7). Suppose that another observation, say Y, is made on the original population. We now post show that the predictive distribution of Y is normal with mean mpost and variance s2 y2 . post The predictive density f *( y) of Y is given by f *( y) f ( y u x)
10 B(x x
1, 11
1 11
0, 1, c, 10
3f (y, uu x)du
3f (y u u) p(uu x) du
1 1 (u mpost)2 (y u)2e du 2s2 2y2post
After some simplification we get
f * (y)
1 2(s2y2 )>(s2 post y2 ) post
(y2 y s2 mpost) post (s2 y2 ) post
1 2(s2y2 )>(s2 post y2 ) post
y2 y s2 mpost post s2 y2 post
y2 y s2 m post post s2 y2 post
T du
The exponent in the second factor may be further simplified to yield
f * ( y) ~ 3 e
1 2(s2y2 )>(s2 post y2 ) post
(y2post y s2 mpost) s2 y2 post
1 (y mpost)2 du 2(s2 y2 ) post
The second factor here is free from u. The first factor is a normal density in u and integrates out to an expression free from u and y. Therefore, the preceding integral becomes e
1 (y mpost)2 2(s2 y2post)
This may be recognized as a normal density with mean mpost and variance s2 y2 . We thus see that predicpost tive density of the future observation Y is normal with mean equal to the posterior mean of u and variance equal to the sum of the population variance and the posterior variance of u. The following theorem is a straightforward generalization of this result (see Problem 11.96). Theorem 11-12 Suppose that a random sample of size n is drawn from a normal distribution with unknown mean u and known variance s2 and that the prior distribution of u is normal with mean m and variance y2. If a second independent sample of size m is drawn from the same population, then the predictive distribution of the sample mean is normal with mean mpost and variance s2m ny2x 2 # s2 s2y2 y2 , where mpost m , ypost , and x is the mean of the first # post 2 2 2 s ny s ny2 sample, of size n.
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