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1 (see page 37). Prior densities that satisfy the first condition but violate the second due
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to the integral being divergent have, however, been employed within the Bayesian framework and are referred to as improper priors. They often arise as natural choices for representing vague prior information about parameters with infinite range. For example, when sampling from a normal population with known mean, say 0, but unknown variance u, we 1 , u 0. Given a sample of observations may assume that the prior density for u is given by p(u) u
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CHAPTER 11 Bayesian Methods
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x (x1, x2, c, xn), if we overlook the fact that the prior is improper and apply formula (1), we get the posterior density a x2 i
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1 p(uu x) ~ n>2 exp u
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This is a proper density, known as an inverse gamma, with parameters a n>2 and b g i x2 >2 (see Problem i 11.99). We have thus arrived at a proper posterior density starting with an improper prior. Indeed, this will be true in all of the situations with improper priors that we encounter here, although this is not always the case.
EXAMPLE 11.9 Suppose that X is binomial with known n and unknown success probability u. The prior density for 1 u given by p(u) , 0 u 1 is improper and is known as Haldane s prior. Let us overlook the fact that p(u) u(1 u) is improper, and proceed formally to derive the posterior density p(u u x) corresponding to an observed value x of X:
p(u u x)
f (x u u)p(u) 3 f (x u u)p(u)du
ux(1 u(1
u)n u)
ux 1(1 u)n x B(x, n x)
We see that the posterior is a proper beta density with parameters x and n x.
EXAMPLE 11.10 Suppose X is normally distributed with unknown mean u and known variance s2. An improper prior u ` . This density may be thought of as representing prior distribution for u in this case is given by p(u) 1, ` ignorance in that intervals of the same length have the same weight regardless of their location on the real line. Given the observation vector x (x1, x2, c, xn), the posterior distribution of u under this prior is given by
a (xi p(u u x) ~ f (xu u)p(u) ~ exp
1 ~ exp e
n(u x)2 # f 2s2
which is normal with mean x and variance s2 >n. #
Conjugate Prior Distributions
Note that Theorems 11-1, 11-2, and 11-3 share an important characteristic in that the prior and posterior densities in each belong to the same family of distributions. Whenever this happens, we say that the family of prior distributions used is conjugate (or closed) with respect to the population density f (x uu). Thus the beta family is conjugate with respect to the binomial distribution (Theorem 11-1), the gamma family is conjugate with respect to the Poisson distribution (Theorem 11-2), and the normal family is conjugate with respect to the normal distribution with known variance (Theorem 11-3). Since p(u u x, y) ~ f ( y u u)p(uu x) whenever x and y are two independent samples from f (x uu), conjugate families make it easier to update prior densities in a sequential manner by just changing the parameters of the family (see Example 11.11). Conjugate families are thus desirable in Bayesian analysis and they exist for most of the commonly encountered distributions. In practice, however, prior distributions are to be chosen on the basis of how well they represent one s prior knowledge and beliefs rather than on mathematical convenience. If, however, a conjugate prior distribution closely approximates an appropriate but otherwise unwieldy prior distribution, then the former naturally is a prudent choice. We now show that the gamma family is conjugate with respect to the exponential distribution. Suppose that X has the exponential density, f (x u u) ue ux, x 0, with unknown u, and that the prior density of u is gamma with parameters a and b. The posterior density of u is then given by p(uu x) ~ f (x u u) p(u) une
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