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CHAPTER 11 Bayesian Methods
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Fig. 11-3
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Sampling From a Normal Population with Known Variance
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Theorem 11-3 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 distribution of u is also normal, with mean mpost and variance y2 given by post mpost s2m s2 ny2x # ny2 y2 post s2 s2y2 ny2 (7)
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The likelihood of the observations is given by f (x u u) 1 1 exp e (x (2p)n>2sn 2s2 ia i 1
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u)2 f n(x # u)2. Using this, and ignor-
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We know from Problem 5.20 (see Method 2) that g(xi u)2 g(xi x)2 # ing multiplicative constants not involving u, we can write the likelihood as f (x u u) ~ exp e Using (2) and the fact that p(u) 1 exp e y 22p n (u 2s2 x)2 f #
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p(uu x) ~ exp e
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Completing the square in the expression in brackets, we get p(uu x) ~ exp e [u (xy2 ms2 >n)>(y2 s2 >n)]2 # f [2(s2 >n)y2]>[y2 (s2 >n)]
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1 n B (u 2 s2
1 (u 2y2
m)2 f , we get the posterior density of u as x)2 # 1 (u y2 m)2 R f
This proves that the posterior density of u is normal with mean and variance given in (7). A comparison of the prior and posterior variances of u in Theorem 11-3 brings out some important facts. It is convenient to do the comparison in terms of the reciprocal of the variance, which is known as the precision of the distribution or random variable. Clearly, the smaller the variance of a distribution, the larger would be its precision. Precision is thus a measure of how concentrated a random variable is or how well we know it. In Theorem 11-3, if we denote the precision of the prior and posterior distributions of u, respectively, by jprior and jpost, then we have jprior 1 y2 and jpost s2 ny2 s2y2 1 y2 n s2 (8)
CHAPTER 11 Bayesian Methods
n may be thought of as the precision of the data (sample mean). If we denote this by jdata, we have s2 the result jpost jprior jdata. That is, the precision of the posterior distribution is the sum of the precisions of the prior and of the data. We can also write the posterior mean, given in (7), in the form The quantity mpost s2m s2 ny2x # 2 ny jprior m jprior jdatax # jdata (9)
This says that the posterior mean is a weighted sum of the prior mean and the data, with weights proportional to the respective precisions. Now suppose that jprior is much less than jdata. Then jpost would be very close to jdata, and mpost would be close to x. In other words, the data would then dominate the prior information, and the posterior distribution would es# sentially be proportional to the likelihood. In any event, as can be seen from (8) and (9), the data would dominate the prior for very large n.
EXAMPLE 11.8 Suppose X is normally distributed with unknown mean u and variance 4 and that p(u) is standard normal. If a sample of size n 10 yields a mean of 0.5, then by Theorem 11-3, p(uu x) is normal with mean 0.36 and variance 0.29. The posterior precision (53.5) is more than three times the prior precision (51), which is evident from the densities shown in Fig. 11-4. The precision of the data is 10>4 2.5, which is reasonably larger than the prior precision of 1; and this is reflected in the posterior mean of 0.36 being closer to x 0.5 than to the prior mean, 0. #
0.7 0.6 0.5 0.4 0.3 0.2 0.1
Fig. 11-4
Improper Prior Distributions
The prior probability density functions p(u) we have seen until now are all proper in the sense that (i) p(u)