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CHAPTER 11 Bayesian Methods
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Improper prior distributions 11.19. An improper prior density for a Poisson mean l is defined by p(l) 1 density in this case is gamma with parameters nx 1 and n . #
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Given the observation vector x, the posterior density of l is p(l u x) ~ e 1 density is of the gamma form with parameters nx 1 and n . #
1, l
nlln x.
0. Show that the posterior
The result follows since this
11.20. Another improper prior density for Poisson mean l is p(l) sity in this case is gamma.
We have p(l u x) ~ e
nlln x
1>l, l
0. Show that the posterior den-
1 ~e l
nlln x 1.
1 The posterior is therefore gamma with parameters nx and n . #
11.21. An improper prior density for the Poisson mean, known as Jeffreys prior for the Poisson, is given by p(l) 1> !l, l 0. Find the posterior density under this prior.
Given the observation vector x, the posterior density of l is p(l u x) ~ e
nlln x
1 1 This is a gamma density with parameters nx and n . # 2 11.22. X is binomial with known n and unknown success probability u. An improper prior density for u, known 1 as Haldane s prior, is given by p(u) , 0 u 1. Find the posterior density of u based on u(1 u) the observation x.
p(uu x) n ux(1 x n u)n
1 ~e !l
nlln x 1 2,
1 u(1 u)
x and b
x, so we get p(u u x)
n ux 1(1 x
x 1,
1. This is a beta density with
ux 1(1 u)n x 1 ,0 B(x, n x) n ux x
11.23. Do Problem 11.22 assuming Jeffreys prior for the binomial, given by p(u)
p(uu x) a x n 1 ux(1 u)n x 1>2 u (1 u)1>2 x 1 1 and b n x . 2 2
1 2 (1
1 2u(1
1 2,
1. This is a beta density with
11.24. Suppose we are sampling from an exponential distribution (page 118) with unknown parameter u, which has the improper prior density p(u) 1>u, u 0. Find the posterior density p(uu x).
p(uu x) ~ un e a n and b
ug xi
1 ~ un 1e u 1> a xi.
ug xi,
0. The posterior density for u is therefore gamma with parameters
11.25. X is normal with unknown mean u and known variance s2. The prior distribution of u is improper and is ` u ` . Determine the posterior density p(uu x). given by p(u) 1,
p(uu x)~ e
1 (x u)2 2s2 a i i
1~ e
n 2s2
(u x)2
. The posterior distribution is thus normal with mean x and variance s2 >n. #
11.26. X is normal with mean 0 and unknown variance u. The variance has the improper prior density p(u) 1> !u, u 0. Find the posterior distribution of u.
p(u u x) ~ 1 e u n>2 ~u ~u
2 ax >2u
1 2u u
,u 0 u 1 2
n 1 x2>2u, R 2 e a
n 1 R 1e ax2>2u, 2
0 and b ax . 2
This is an inverse gamma density (see Problem 11.99) with a
CHAPTER 11 Bayesian Methods
Conjugate prior distributions 11.27. A poll to predict the fate of a forthcoming referendum found that 1010 out of 2000 people surveyed were in favor of the referendum. Assuming a prior uniform density for the unknown population proportion u, find the chances that the referendum would lose. Comment on your result with reference to Problems 11.11 and 11.12.
The posterior distribution of u, given the poll result, is beta with parameters 1011, 991. We need the probability that u 0.5. This comes out to be 0.33, so we can be 33% sure that the referendum would lose. This is the same result that we obtained in Problem 11.12 using for prior the posterior beta distribution derived in Problem 11.11. Since the beta family is conjugate with respect to the binomial distribution, we are able to update the posterior sequentially in Problem 11.12.
11.28. A random sample of size 10 drawn from a geometric distribution with success probability u (see page 117) yields a mean of 4.2. The prior density of u is uniform in the interval [0, 1]. Determine the posterior distribution of u. The prior distribution is beta with parameters a b 1. We know from Example 11.12 that the posterior distribution is also beta. The parameters are given by a n 11 and b nx n 33. #
11.29. A random sample of size n is drawn from a negative binomial distribution with parameter u (see page 117): f (x; u) x r u)x r, x r, r 1, c . Suppose that the prior density of u is beta with
parameters a and b. Show that the posterior density of u is also a beta, with parameters a nr and b nx nr, where x is the sample mean. In other words, show that the beta family is conjugate with # # respect to the negative binomial distribution.
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