ssrs 2012 barcode font Bayesian Methods in Software

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CHAPTER 11 Bayesian Methods
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Let D denote the event (data) that two heads and two tails are obtained in four tosses. We then have, from Bayes theorem, P(D u F)P(F) P(D u F)P(F) P(D u B)P(B) 1 P(F u D) 4 1 B (0.5)4 R 2 2 4 1 B (0.5)4 R 2 2 4 1 B (0.3)2(0.7)2 R 2 2
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P(F u D)
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625 < 0.59 1066
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11.3. Verify the posterior probability values given in Table 11-2.
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P(D u u 0.2)P(u 0.2) 0.2)P(u 0.2) P(Du u 0.5)P(u
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0.2u D)
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0.5)
32 < 0.20 157 P(u 0.5u D) 1 P(u 0.2 u D) 125 < 0.80 157
1 [3(0.2)2(0.8)] 2
1 [3(0.2)2(0.8)] 2
1 [3(0.5)3] 2
11.4. The random variable X has a Poisson distribution with an unknown parameter l. It has been determined that l has the subjective prior probability function given in Table 11-6. A random sample of size 2 yields the X-values 2 and 0. Find the posterior distribution of l.
Table 11-6 l p(l) The likelihood of the data is f (x ul) p(l u x) ~ e e
0.5 1>2
1.0 1>3
1.5 1>6
lx1 x2 . The posterior density is (up to factors free from l) x1!x2! ~e
2ll2p(l)
2llx1 x2p(l)
0.5, 1, 1.5
The results are summarized in Table 11.7. Table 11-7 l p(l u x) 0.5 0.42 1.0 0.41 1.5 0.17
11.5. In a lot of n bolts produced by a machine, an unknown number r are defective. Assume that r has a prior binomial distribution with parameter p. Find the posterior distribution of r if a bolt chosen at random from the lot is (a) defective; (b) not defective.
(a) We are given the prior probability function p(r) probability function of r, given the event D n r r n defective, is p(r uD) ~ n pr(1 r p)n r, r 1, c, n n pr(1 r p)n r, r 0, 1, c, n. The posterior p)n r, r 0,1, c, n
1 r 1 p (1 p)n r 1, the constant of proportionality in the preceding probability 1 n 1 r 1 1 function must be p . Therefore, p(r uD) p (1 p)n r, r 1, c, n. r 1
n n Since a r r 1
1 r p (1 1
n pr(1 r pr(1 p
CHAPTER 11 Bayesian Methods
n n n r 1 r r
(b) p(ru Dr) ~
p)n r, r
0, 1, c, n 1
p)n r, r
0, c, n
Since a
n 1 r 0
function must be
pr(1 1
1, the constant of proportionality in the preceding probability n r 1 pr(1 p)n
1 r,
. Therefore, p(r u Dr)
0, c, n
11.6. X is a binomial random variable with known n and unknown success probability u. Find the posterior density of u assuming a prior density p(u) equal to (a) 2u, 0 u 1; (b) 3u2, 0 u 1; (c) 4u3, 0 u 1.
(a) p(uu x) ~ ux(1 u)n
ux 1(1
u)n x, 0
u 2 and n
1. x x x x 1) 1) 1, the normalizing constant is 1) ux 1(1 u)n x, 0 u)n x, 0 u)n x, 0 u u u 1. 1. 1. 1 2, n
Since this is a beta density with parameters x 1>B(x 2, n x 1) and we get p(uu x)
B(x B(x B(x
(b) The posterior is the beta density: p(u u x) (c) The posterior is the beta density: p(u u x)
1 3, n 1 4, n
ux 2(1 ux 3(1
11.7. A random sample x f (xu u) 3ux2 e ux3, x sity of u.
p(uu x) ~ une n a and 1
uax3
(x1, x2, c, xn) of size n is taken from a population with density function 0. u has a prior gamma density with parameters a and b. Find the posterior den1 R b .
ua 1e x3
b ba
posterior density is p(u u x)
. The normalizing constant should therefore be
~ un
a 1e u Qax3
This may be recognized as a gamma density with parameters 1 b a x3 n b
1 R b ,
11.8. X is normal with mean 0 and unknown precision j which has prior gamma density with parameters a and b . Find the posterior distribution of j based on a random sample x (x1, x2, c, xn) from X.
p(jux) ~ jn>2 e
2a j x2
ja 1 e
~ j2
a 1 e jQ
ax 2
b a x3 n a n u b
1 b R,
1 (n a)
and the
a 1e u Qax3
j n 2
0 a and 2b b a x2 2 .
Therefore, j has a gamma distribution with parameters
Sampling from a binomial population 11.9. A poll of 100 voters chosen at random from all voters in a given district indicated that 55% of them were in favor of a particular candidate. Suppose that prior to the poll we believe that the true proportion u of voters in that district favoring that candidate has a uniform density over the interval [0, 1]. Find the posterior density of u.
Applying Theorem 11-1 with n and b 46. 100 and x 55, the posterior density of u is beta with parameters a 56
11.10. In 40 tosses of a coin, 24 heads were obtained. Find the posterior distribution of the proportion u of heads that would be obtained in an unlimited number of tosses of the coin. Use a uniform prior for u.
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