Hi all, does anyone know the following proofs?
Suppose that you have a random sample x1, x2, . . . , xn from an N(μ,1/r) distri-
bution (with r known) and take a N(b,1/d) prior for μ. Verify that:
(i) the posterior mean is greater than the prior mean if and only if the likelihood
mode is greater than the prior mean;
(ii) the standard deviation of the posterior distribution is smaller than that of
the prior distribution.
Thanks!