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The posterior density, given by p(u u x)~f (x u u) p(u) ~ u is inverse gamma with parameters na g and b
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Bayesian point estimation 11.37. In Problem 11.5, find the Bayes estimate of r with squared error loss function.
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(a) The Bayes estimate is the mean of the posterior distribution, which is n n n 1 r 1 n 1 r 1 r p (1 p)n r 1 a a (r 1) r 1 p (1 r 1 r 1 r 1 (b) The posterior mean is a r
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CHAPTER 11 Bayesian Methods
11.38. Show that the Bayes estimate mpost for u obtained in Theorem 11-5 is a convex combination of the maximum likelihood estimate of u and the prior mean of u.
mpost x n a a b n n a b x n a n a b b a a b
11.39. In Problem 11.10, find the Bayes estimate with squared error loss function for (a) u (b) 1>u.
(a) The posterior distribution is beta with parameters 25 and 17. The Bayes estimate, which equals the posterior mean, is 25>52 < 0.48. (b) The Bayes estimate of 1>u is the posterior mean of 1>u, given by
1 1 u24(1 B(25,17) 3 u
u)16 du
B(24, 17) B(25, 17)
41 < 1.71 24
11.40. In Problem 11.15, find the Bayes estimate with squared error loss function for u.
The Bayes estimate is the posterior mean, which we know from Problem 11.15 to be 4.38.
11.41. In Problem 11.33, assume that a
1 and find the Bayes estimate for the variance with squared error loss.
a xi 1 and 1
n The posterior distribution is inverse gamma (see Problem 11.99) with parameters 2 2 2 a xi i Bayes estimate is the posterior mean, given by . n
. The
11.42. Find the Bayes estimate of u with squared error loss function in Problem 11.24 and compare it to the maximum likelihood estimate.
The parameters of the posterior are n and 1^ a xi. Therefore, the Bayes estimate, which is the posterior mean,
is 1>x. This is the same as the maximum likelihood estimate for u (see Problem 11.98). #
11.43. In Example 11.10, determine the Bayes estimate for u under the squared error loss function.
The posterior distribution of u is normal with mean x and variance s2 >n. The Bayes estimate of u under the # squared error loss, which is the posterior mean, is given by x. #
11.44. In Problem 11.30, find the Bayes estimate for u with squared error loss function. Find the squared error loss (x1, x2, c, xn ) and compare it to the loss of the maximum likelihood estimate. of the estimate for each x
The Bayes estimate under squared error loss is the posterior mean b(a n)>(1 nbx). The squared error # loss for each x is the posterior variance b2(a n)>(1 nbx)2. With a 1 and b 0.2, n 11 and x 6, # # this comes to 0.00238. The maximum likelihood estimate for u is 1>x and its squared error loss is # 1 EB x # u 2 xR
to 0.00239.
11.45. If X is a Poisson random variable with parameter l and the prior density of l is gamma with parameters a and b, then show that the Bayes estimate for l is a weighted average of its maximum likelihood estimate and the prior mean.
By Theorem 11-2, the posterior distribution is gamma with parameters nx # b(nx a) nb # 1 mean is x ab. (1 nb) 1 nb # 1 nb a and b>(1 nb). The posterior
1 2 x #
2 x E(u u x) #
E(u2 u x). With a
1 and b
0.2, n
11 and x #
6, this comes
11.46. In Problem 11.16, find the Bayes estimate of u with (a) squared error loss and (b) absolute error loss.
(a) The Bayes estimate with squared error loss is the posterior mean of u, which is 1.17. (b) The Bayes estimate with absolute error loss is the posterior median, which is the same as the posterior mean in this case since the posterior distribution is normal.
CHAPTER 11 Bayesian Methods
11.47. In Problem 11.32, find the Bayes estimate for u with squared error loss function.
The posterior distribution of u is gamma with parameters a 2b(a n) mean is . Q2 b a x2 R i
n and
2b 2 b a x2
. Therefore, the posterior
11.48. The time (in minutes) that a bank customer has to wait in line to be served is exponentially distributed with mean 1>u. The prior distribution of u is gamma with mean 0.4 and standard deviation 1. The following waiting times were recorded for a random sample of 10 customers: 2, 3.5, 1, 5, 4.5, 3, 2.5, 1, 1.5, 1. Find the Bayes estimate for u with (a) squared error and (b) absolute error loss function.
The gamma distribution with parameters a and b has mean ab and variance ab2. Therefore, the parameters for our gamma prior must be a 0.16 and b 2.5. The posterior distribution is (see Theorem 11-4) gamma with parameters a n 10.16 and b>(1 nb x 0.04). # (a) The posterior mean is 10.16 0.04 0.41.
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