4. Let G(0.5)

k (t) be the posterior median of Gk(t).

(a) Show that for each t, G(0.5)

k (t) minimizes expected, integrated absolute loss, E 8 | Gest k (t) − Gk(t) | dt9

.

(b) Outline an algorithm to compute G(0.5)

k (t) using MCMC output.

(c) Let k = 100 and assume that, a posteriori (θ1,...,θk) are i.i.d. with distribution that is a 50/50 mixture of a N(–2, 1) and a N(2,1). Simulate from this distribution by drawing B = 1000 samples each of size 100 and compute G¯k and G(0.5)

k (t) from the output. Graphically and numerically compare these estimates to each other and to the generating distribution.