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Gertjan Verhoeven pointed out a bug in exercise 5. That was based on exercise 5.9(b) of BDA 3 (p. 136). But the phrasing was botched int he text and exercise. It should be as follows.
BDA 3, Problem 5.9(b):
Assume p(alpha / (alpha + beta), alpha + beta) is uniform and show that
p(alpha, beta) propto (alpha + beta)^{-5/2}.
First note that alpha, beta > 0.
Step 1. Let
f(kappa) = kappa^{-0.5} = gamma
so that
f^{-1}(gamma) = gamma^{-2} = kappa
and assume a uniform so that that p_G(gamma) = const. Then with the usual change of variables, we get
Gertjan Verhoeven pointed out a bug in exercise 5. That was based on exercise 5.9(b) of BDA 3 (p. 136). But the phrasing was botched int he text and exercise. It should be as follows.
BDA 3, Problem 5.9(b):
Assume
p(alpha / (alpha + beta), alpha + beta)
is uniform and show thatFirst note that
alpha, beta > 0
.Step 1. Let
so that
and assume a uniform so that that
p_G(gamma) = const
. Then with the usual change of variables, we getStep 2. Compute the Jacobian of the inverse transform
h
, defined bythe Jacobian of which is
which using the rule for determinants for 2 x 2 matrices, provide an absolute Jacobian determinant of:
|J_h| = (alpha + beta)^{-1}
Step 3. Multiply the distribution from Steps 1 and 2 by the absolute Jacobian determinant to adjust for change of variables, and the final result is
Or you could just put the two bits together and derive the compound Jacobian, but it seems clearer this way in two steps.
fix text motivation for prior
fix exercise and give a big hint
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