Incorporating Risk Aversion in Bayesian Expected Loss functions In Berger's Statistical Decision Theory and Bayesian Analysis, he presents the following expected loss function for decision theory:
$\rho(\pi^*,a)=\int_\Theta L(\theta,a)d\pi^*(\theta)$
Where $\theta$ is the state of nature from the set of possible states of nature $\Theta$, $a$ is the action chosen from the set of possible actions $\mathscr{A}$, and $\pi^*$ is the posterior distribution.
Question: how would I incorporate risk aversion into this function? In the above formulation, the loss function would be identical for a certain loss of 50 or a {0,100} space with 50/50 chance of either.
 A: The concept of risk aversion enters into analysis when you have a measure of objective gain/loss that is a random variable, and you wish to obtain an overall measure of subjective gain/loss that accounts for the full distribution of this random variable.  This occurs commonly in the context of economics where the objective gain/loss is a random amount of wealth, income, goods, etc., and the subjective measure is the expected utility corresponding to this.  Within this context, risk aversion occurs when the utility is a concave function of the objective measure.
So, with this in mind, it makes sense to interpret the loss function as measuring the objective loss under some (random) state of nature $\theta \in \Theta$ and decision $a \in \mathscr{A}$.  The posterior expected utility under any action $a$ would be written as:
$$\bar{u}(a) \equiv \mathbb{E}(u(-L(\theta, a))) = \int \limits_\Theta u(-L(\theta, a)) d\pi^*(\theta),$$
where $\pi^*$ is the posterior distribution for $\theta$, and $U: \mathbb{R} \rightarrow \mathbb{R}$ is a subjective utility function satisfying all the normal requirements of utility theory (i.e., it is non-decreasing, etc.).  Assuming that this utility is twice-differentiable, risk aversion occurs in the case where $u'' < 0$ so that utility is concave.
The most commonly used form of utility function to represent risk-aversion is the CES utility with constant-relative-risk-aversion parameter $\rho > 0$.  This utility function is given by:
$$u(c) = \begin{cases} 
      \ln c & \rho = 1 \\[8pt]
      \frac{c^{1-\rho}-1}{1-\rho} & \rho \neq 1 
   \end{cases}$$
In the text you are looking at by Berger, his expression is showing the expected loss, and so he is not including any risk aversion.  Minimisation of expected loss is optimal in the case of a risk-neutral utility function.  If you want to add risk-aversion, you simply transform the negative loss (i.e., the gain) through any concave utility function, whether that be the CES function or another one.
