.
Suppose the data (x1, . . . , x7) = (6.52, 8.32, 0.31, 2.82, 9.96, 0.14, 9.64) are observed.
Consider Bayesian estimation of
μ
based on a
N(μ,
32/7) likelihood for the minimally sufficient ¯x
|
μ, and a Cauchy(5,2) prior.
a.
Using a numerical integration method of your choice, show that the proportionality constant is roughly 7.84654. (In other words, find
k
such that
_
k
× (prior) ×
(likelihood)
dμ
= 1.)
b.
Using the value 7.84654 from (a), determine the posterior probability that 2 ≤
μ
≤ 8 using the Riemann, trapezoidal, and Simpson’s rules over the range of integration [implementing Simpson’s rule as in (5.20) by pairing adjacent subintervals]. Compute the estimates until relative convergence within 0.0001 is achieved for the slowest method. Table the results. How close are your estimates to the correct answer of 0.99605?
c.
Find the posterior probability that
μ
≥ 3 in the following two ways. Since the range of integration is infinite, use the transformation
u
= exp{μ}/(1 + exp{μ}). First, ignore the singularity at 1 and find the value of the integral using one or more
quadrature methods. Second, fix the singularity at 1 using one or more appropriate strategies, and find the value of the integral. Compare your results. How close are the estimates to the correct answer of 0.99086?
d.
Use the transformation
u
= 1/μ, and obtain a good estimate for the integral in part (c).