Question: Exercise 3.1 Show that (a) the posterior mean minimizes expression (3.3), (b) the posterior median minimizes expression (3.3) whereby the square loss function is replaced

Exercise 3.1 Show that

(a) the posterior mean minimizes expression (3.3),

(b) the posterior median minimizes expression (3.3) whereby the square loss function is replaced by a|θ − θ| with a > 0, and

(c) the posterior mode minimizes expression (3.3) with a penalty of 1 if we choose the incorrect value and no penalty for the correct value.

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