4.20 This exercise is related to Example 4.16.

(i) Show that in this case the exact value of (4.62) is given by

√

νπΓ ( ν

2 )

Γ (ν+1 2 )

, and the Laplace approximation (4.63) is

, 2νπ

ν + 1

.

(ii) Show that if q(x) = (x − μ)2/2σ2 for some μ ∈ R and σ2 > 0, the Laplace approximation (4.63) is exact.