7.31 (Millar, 2004.) Consider the case of a random variable X which has a N(, 2) distribution,...

7.31 (Millar, 2004.) Consider the case of a random variable X which has a N(μ, σ2) distribution, where σ2 is known, and the parameter μ has a N(ν, τ 2) distribution.

An important issue in Bayesian analysis is the sensitivity of posterior distributions to assumptions made regarding prior distributions. Let the posterior distribution for μ have mean ν∗. One way to evaluate the sensitivity of the posterior distribution to the prior is to form the derivative, ∂ν∗

∂ν .

Show for the example of this question that the above derivative is the ratio of posterior variance to prior variance, and discuss whether you think this is a sensible result.