1. Decision Theory. In this book, we usually use the posterior mean as a point estimate. However, in a formal decision theoretic context, the choice of a point estimate of
is made by defining a loss function and choosing the point estimate which minimizes expected loss. Thus, if C.e
;
/ is the loss (or cost)

associated with choosinge
as a point estimate of
, then we would choosee

which minimizes E[C.e
;
/jy] (where the expectation is taken with respect to the posterior of
). For the case where
is a scalar, show the following:

(a) Squared error loss function. If C.e
;
/ D .e

/2 thene
D E.
jy/.

(b) Asymmetric linear loss function. If C.e
;
/ D

(

c1je

j ife


c2je

j if e
>

where c1 > 0 and c2 > 0 are constants, thene
is the c1 c1Cc2 th quantile of p.
jy/.

(c) All-or-nothing loss function. If C.e
;
/ D

(

c ife
6D

0 ife
D

where c > 0 is a constant, thene
is the mode of p.
jy/.