Suppose that X= (X,, X) is a sample from a N(0, o2) distribution, where both and o2 are unknown. The prior density of and o² is

π(θ, σ²) = πι(θσ²)π,(σ²), where π, (θ|σ²) is a Ν(μ, τσ²) density and π2(σ²) is an %(α, β) density.

(a) Show that the joint posterior density of and o² given x is

п(6, 02x) = п (00, x)7π2(0²x)

where (6 o2, x) isanormal density with mean μ(x) = (μ + птx)/(пт+1)(here x=(1/n) Σ", x₁) and variance (r¹+n)6², and 7₂(σ²x) is an inverted gamma density with parameters a + n/2 and ẞ', where B 2

(b) Show that the marginal posterior density of o2 given x is I G ( a + n/2, B').(To find the marginal posterior density of o² given x, just integrate out over e in the joint posterior density.)

(c) Show that the marginal posterior density of given x is a density.

T(2a+n, μ(х), [(т1+n) (a+n/2)B']1)

(d) State why the joint prior density in this problem is from a conjugate familyfor the distribution of X. (Note that, for this prior, the conditional priorvariance of given o2 is proportional to 2. This is unattractive, in that prior knowledge of should generally not depend on the sample variance.

If, for example, the X, are measurements of some real quantity 0, it seems silly to base prior beliefs about 0 on the accuracy of the measuring instrument. It must be admitted that the robust prior used in Subsection 4.7.10 suffers from a similar intuitive inadequacy. The excellent resultsobtained from its use, however, indicate that the above intuitive objection is not necessarily damning.)