16. Study the frequentist and Bayesian coverage probabilities for Bayesian intervals based on the Gaussian/Gaussian model with a N(μ, τ 2) prior and a sampling distribution, conditional on θ, that is N(θ, σ2).

(a) Show you can assume μ = 0 and σ2 = 1 without loss of generality .

(b) Show that the highest posterior probability interval is also equaltailed.

(c) Evaluate the frequentist coverage of the Bayesian interval for combinations of the prior variance, τ 2, and the (fixed) parameter of interest,

θ. For example, plot coverage versus θ for several values of τ 2.

(d) Evaluate the Bayesian posterior coverage and preposterior coverage of the interval based on μ = 0, when the true prior mean is not 0.

(Hint: All probabilities can be represented as Gaussian integrals, and the interval endpoints come directly from the Gaussian cumulative distribution function.)